#361003

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1231 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)(3,6)(4,5)(7,8)(9,10,14,21,15,19,13,20)(11,22,26,33,27,16,25,34)(12,17,28,36,29,18,30,37)(23,38,24,32,39,31,40,35);(9,11,15,27)(10,16,19,22)(12,24,29,40)(13,26,14,25)(17,35,18,32)(20,34,21,33)(23,30,39,28)(31,37,38,36);(2,5)(6,8)(9,12)(10,17)(11,23)(13,30)(14,28)(15,29)(16,31)(18,19)(20,37)(21,36)(22,38)(24,26)(25,40)(27,39)(32,33)(34,35);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,24)(13,28)(14,30)(15,29)(16,32)(17,19)(20,37)(21,36)(22,35)(23,26)(25,39)(27,40)(31,33)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268199);