#6545337

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
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	C₂²×C₄	C₂⁵
prime radical	#61164	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(9,10,14,18,15,19,13,28)(11,20,24,31,22,35,23,16)(12,25,26,33,27,32,29,17)(21,30,37,39,36,40,38,34);(1,2)(3,6)(4,5)(7,8)(9,11,14,24,15,22,13,23)(10,16,28,35,19,31,18,20)(12,21,29,38,27,36,26,37)(17,30,25,39,33,40,32,34);(2,5)(6,8)(9,12,15,27)(10,17,19,33)(11,21,22,36)(13,26,14,29)(16,30,31,40)(18,32,28,25)(20,34,35,39)(23,37,24,38);(1,3)(2,6)(4,7)(5,8)(11,22)(16,31)(20,35)(21,36)(23,24)(30,40)(34,39)(37,38);(3,7)(6,8)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,g^c=gi,e^d=eh

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, 6452533);