#359098

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1372 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)(5,6,8,7)(9,10)(11,19)(12,23)(13,27)(14,17)(15,21)(16,26)(18,29)(20,36)(22,34)(24,35)(25,33)(28,31)(30,39)(32,37)(38,40);(9,11,14,22)(10,15,17,31)(12,24,26,20)(13,28,29,21)(16,32,23,30)(18,34,27,19)(25,37,40,39)(33,35,38,36);(2,4)(9,12,13,25,14,26,29,40)(10,16,18,33,17,23,27,38)(11,20,21,37,22,24,28,39)(15,30,19,35,31,32,34,36);(5,7,8,6)(10,17)(15,31)(16,23)(18,27)(19,34)(30,32)(33,38)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=cg,e^b=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266294);