#7500804

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3214 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408000);