#364483

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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

2474 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=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, 271679);