#6364110

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,18,23,26)(19,29,30,31)(20,25,24,32)(21,28,22,27);(2,7)(3,5)(4,11)(8,10)(9,16)(12,13)(17,19)(18,24)(20,26)(21,32)(22,25)(23,30)(27,31)(28,29);(1,3,6,5)(2,8,10,7)(4,11,12,13)(9,15,16,14)(17,20)(18,19)(21,31)(22,29)(23,24)(25,27)(26,30)(28,32);(1,4,6,12)(2,9,10,16)(3,11,5,13)(7,14,8,15)(17,20)(18,25)(19,27)(21,29)(22,31)(23,24)(26,32)(28,30);(2,10)(7,8)(9,16)(14,15)(18,26)(19,30)(25,32)(27,28)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283124, #72283862, #72283969, …
maximal normal subgroup of	#72283124, #72283862, #72283969, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271306);