#450926

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358122);