#232165

order	512
nilpotency class	3
isoclinism	isoclinic class of #232153
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

425 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efhi,d^2=eh,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cef,d^b=de,e^b=ei,f^b=fhi,g^b=ghi,g^c=gh,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72245104
maximal subgroup of	#72245104, #72245124
maximal normal subgroup of	#72245104, #72245124

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139361);