#434826

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342022);