#375878

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283074);