#397959

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1973 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)(11,20)(12,17)(13,27)(14,18)(15,19)(16,22)(21,33)(23,31)(24,39)(25,34)(26,36)(28,32)(29,35)(30,38)(37,40);(9,11,13,22,15,23,14,28)(10,16,18,20,19,32,27,31)(12,24,26,21,25,38,29,37)(17,33,35,39,34,40,36,30);(1,2,3,4)(9,12,15,25)(10,17,19,34)(11,21,23,37)(13,26,14,29)(16,30,32,39)(18,35,27,36)(20,33,31,40)(22,38,28,24);(6,8)(12,25)(17,34)(21,37)(24,38)(26,29)(30,39)(33,40)(35,36)

Polycyclic presentation:

2^9|b^2=e,c^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=di,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, 305155);