#201065

order	512
nilpotency class	3
isoclinism	isoclinic class of #200938
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

991 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 991 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108261);