#357467

order	512
nilpotency class	3
isoclinism	isoclinic class of #184767
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4342 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)(3,4)(9,10,13,18,14,17,28,32)(11,19,22,33,21,15,26,30)(12,23,24,34,25,16,27,31)(20,29,37,39,36,35,38,40);(2,4)(5,6)(7,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,29)(17,19)(18,33)(22,28)(23,35)(24,38)(25,36)(27,37)(30,32)(31,40)(34,39);(6,8)(9,12,14,25)(10,16,17,23)(11,20,21,36)(13,27,28,24)(15,29,19,35)(18,34,32,31)(22,38,26,37)(30,40,33,39);(11,21)(15,19)(20,36)(22,26)(29,35)(30,33)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,h^b=hi,h^c=hi

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, 264663);