#373699

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2222 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)(11,16)(12,24)(13,18)(14,29)(15,20)(17,27)(19,22)(21,34)(23,33)(25,40)(26,36)(28,32)(30,35)(31,38)(37,39);(1,2)(3,4)(9,11,15,23)(10,16,20,33)(12,21,26,37)(13,14,28,22)(17,31,35,40)(18,29,32,19)(24,39,36,34)(25,30,38,27);(2,4)(5,7,8,6)(9,12)(10,17)(11,21)(13,25)(14,30)(15,26)(16,31)(18,34)(19,36)(20,35)(22,27)(23,37)(24,29)(28,38)(32,39)(33,40);(9,13,15,28)(10,18,20,32)(11,22,23,14)(12,25,26,38)(16,19,33,29)(17,34,35,39)(21,27,37,30)(24,31,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,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, 280895);