#6360879

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360760
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

1590 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)(3,9,12,15)(4,13,14,8)(6,10,16,11)(17,18)(19,23)(20,27)(21,30)(22,25)(24,26)(28,32)(29,31);(1,3,5,12)(2,6,7,16)(4,10,14,11)(8,15,13,9)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,16)(13,14)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12);(2,7)(6,16)(8,13)(9,15)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,31,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283964, #72338420, #72338431
maximal normal subgroup of	#72283964, #72338420, #72338431

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6268075);