#187745

order	512
nilpotency class	3
isoclinism	isoclinic class of #187721
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62840	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2028 conjugacy classes of subgroups; only the 304 normal subgroups are shown.

normal subgroups only

This group has 2028 conjugacy classes of subgroups — too many to draw. Only its 304 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,14,9,3,16,21)(4,23,26,56,28,37,48,55)(5,13,29,47,30,20,49,42)(7,18,31,45,34,11,41,43)(8,15,32,12,35,22,50,19)(10,36,39,58,17,51,38,24)(25,59,27,46,60,64,61,54)(33,40,62,44,63,52,57,53);(1,3)(2,9)(4,17)(5,13)(6,12)(7,18)(8,22)(10,28)(11,34)(14,32)(15,35)(16,19)(20,30)(21,50)(23,33)(24,46)(25,38)(26,44)(27,52)(29,43)(31,42)(36,55)(37,63)(39,60)(40,61)(41,47)(45,49)(48,53)(51,56)(54,58)(57,64)(59,62);(1,4,8,27)(2,10,15,40)(3,17,22,52)(5,24,7,33)(6,26,32,60)(9,28,35,61)(11,37,20,54)(12,44,14,39)(13,46,18,23)(16,48,50,25)(19,53,21,38)(29,36,31,62)(30,58,34,63)(41,57,49,51)(42,59,43,55)(45,56,47,64);(1,5,8,7)(2,11,15,20)(3,18,22,13)(4,24,27,33)(6,29,32,31)(9,30,35,34)(10,37,40,54)(12,42,14,43)(16,49,50,41)(17,23,52,46)(19,47,21,45)(25,57,48,51)(26,36,60,62)(28,58,61,63)(38,56,53,64)(39,55,44,59)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72272599
maximal subgroup of	#72272561, #72272564, #72272570, …
maximal normal subgroup of	#72272561, #72272564, #72272570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94941);