#412336

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 319532);