#6355518

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60559	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,22)(19,26)(20,29)(24,25)(27,28)(30,32);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,18,20,22,21,24,29,25)(19,23,27,30,28,31,26,32);(17,19)(18,23)(20,27)(21,28)(22,30)(24,31)(25,32)(26,29);(18,24)(22,25)(23,31)(30,32);(3,10)(6,15)(9,16)(11,14)(17,20,21,29)(18,25,24,22)(19,27,28,26)(23,32,31,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306856, #72306906
maximal normal subgroup of	#72306856, #72306906

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262714);