#355943

order	512
nilpotency class	2
isoclinism	isoclinic class of #180872
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,h^a=hi,c^b=ce,d^b=di,g^c=gi

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, 263139);