#373653

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2216 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)(9,10,14,18,15,19,13,27)(11,16,23,31,21,30,22,37)(12,24,25,17,26,32,28,33)(20,34,36,29,35,40,38,39);(2,4)(5,6)(7,8)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,29)(18,31)(19,30)(24,34)(25,36)(26,35)(27,37)(28,38)(32,40)(33,39);(6,8)(9,12,14,25,15,26,13,28)(10,17,18,32,19,33,27,24)(11,20,23,36,21,35,22,38)(16,29,31,40,30,39,37,34);(2,4)(11,21)(16,30)(20,35)(22,23)(29,39)(31,37)(34,40)(36,38)

Polycyclic presentation:

2^9|a^2=h,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,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, 280849);