#6367094

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

1604 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,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	#72306848, #72306850, #72306874
maximal normal subgroup of	#72306848, #72306850, #72306874

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274290);