#429934

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1001 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:

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

Polycyclic presentation:

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

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, 337130);