#6360683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360376
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

1553 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283574
maximal normal subgroup of	#72283574

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6267879);