#6372762

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34633	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2830 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279958);