#6364099

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

1633 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=i,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	#72306813, #72306857, #72306861
maximal normal subgroup of	#72306813, #72306857, #72306861

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271295);