#360819

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1239 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)(3,7)(4,6)(5,8)(9,10,13,18,14,17,25,31)(11,15,21,28,20,27,34,38)(12,16,24,29,23,30,36,39)(19,32,33,40,22,26,35,37);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,22)(13,21)(14,20)(16,26)(17,27)(18,28)(19,23)(24,35)(25,34)(29,37)(30,32)(31,38)(33,36)(39,40);(1,3,5,4)(2,7,8,6)(9,12)(10,16)(11,19,20,22)(13,24)(14,23)(15,26,27,32)(17,30)(18,29)(21,33,34,35)(25,36)(28,37,38,40)(31,39);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268015);