#423830

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=f,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi

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, 331026);