#6367241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
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

1651 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307001, #72307008, #72307025
maximal normal subgroup of	#72307001, #72307008, #72307025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274437);