#360209

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1664 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:

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

Polycyclic presentation:

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

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, 267405);