#6367171

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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

1570 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,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	#72306813, #72306848, #72306853
maximal normal subgroup of	#72306813, #72306848, #72306853

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274367);