#6352296

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352157
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35840	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259492);