#6368874

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
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

1509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

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=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306706, #72306707, #72306725
maximal normal subgroup of	#72306706, #72306707, #72306725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276070);