#391977

order	512
nilpotency class	3
isoclinism	isoclinic class of #232147
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

856 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)(3,5)(4,6)(7,8)(9,10,14,19,13,18,15,29)(11,16,22,32,23,33,21,31)(12,24,27,35,26,36,28,17)(20,34,25,39,38,40,37,30);(1,3)(2,5)(4,7)(6,8)(9,11,15,21,13,23,14,22)(10,16,19,32,18,33,29,31)(12,25,28,20,26,37,27,38)(17,34,24,39,35,40,36,30);(2,6)(5,8)(9,12)(10,17)(11,20)(13,26)(14,28)(15,27)(16,30)(18,35)(19,36)(21,25)(22,37)(23,38)(24,29)(31,34)(32,40)(33,39);(3,7)(5,8)(11,21,23,22)(12,26)(16,31,33,32)(17,35)(20,25,38,37)(24,36)(27,28)(30,34,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299173);