#365056

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1642 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=ei,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, 272252);