#353779

order	512
nilpotency class	2
isoclinism	isoclinic class of #180864
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×Q₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1650 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)(3,6)(4,5)(7,8)(9,10,13,18,15,19,14,28)(11,20,23,16,24,30,22,31)(12,17,26,33,25,32,27,38)(21,34,37,29,35,40,36,39);(9,11,14,23,15,24,13,22)(10,16,18,20,19,31,28,30)(12,21,27,37,25,35,26,36)(17,29,33,34,32,39,38,40);(2,5)(6,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,33)(19,32)(20,34)(22,36)(23,37)(24,35)(28,38)(30,40)(31,39);(1,3,4,7)(2,6,5,8)(12,25)(17,32)(21,35)(26,27)(29,39)(33,38)(34,40)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260975);