#429275

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

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

875 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)(3,6)(4,8)(5,7)(9,10,13,18,14,19,15,20)(11,21,23,32,24,33,25,16)(12,26,28,36,29,17,30,35)(22,37,27,31,38,34,39,40);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,16,19,32)(12,27,29,39)(13,25,15,23)(17,34,26,37)(18,33,20,21)(22,30,38,28)(31,36,40,35);(2,7)(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,35)(19,26)(20,36)(21,34)(23,27)(24,38)(25,39)(32,40)(33,37);(1,4,5,3)(2,8,7,6)(9,12)(10,17)(11,22)(13,30)(14,29)(15,28)(16,31)(18,36)(19,26)(20,35)(21,37)(23,39)(24,38)(25,27)(32,40)(33,34)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=fi,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, 336471);