#356036

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#376
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

810 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=h,d^2=e,f^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, 263232);