#360120

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1644 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=ei,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, 267316);