#376814

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1007 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)(9,10,15,19,14,18,30,34)(11,20,24,36,23,16,31,33)(12,17,27,13,26,29,40,28)(21,32,37,22,25,35,39,38);(1,3,4,8)(2,5,6,7)(9,11,14,23)(10,16,18,20)(12,25,26,21)(13,22,28,38)(15,24,30,31)(17,32,29,35)(19,33,34,36)(27,39,40,37);(9,12)(10,17)(11,21,23,25)(13,19)(14,26)(15,27)(16,32,20,35)(18,29)(22,36,38,33)(24,37,31,39)(28,34)(30,40);(2,6)(5,7)(9,13,15,29,14,28,30,17)(10,12,19,27,18,26,34,40)(11,22,24,35,23,38,31,32)(16,25,33,39,20,21,36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=h,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, 284010);