#394873

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1396 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)(3,8,4,6)(9,10,14,19)(11,21,13,17)(12,24,27,37)(15,31,16,20)(18,29,36,28)(22,35,26,40)(23,33,30,34)(25,38,39,32);(1,3,5,4)(2,6,7,8)(9,11,14,13)(10,17,19,21)(12,25,27,39)(15,23,16,30)(18,35,36,40)(20,33,31,34)(22,28,26,29)(24,32,37,38);(1,4,5,3)(2,6,7,8)(9,12,14,27)(10,18,19,36)(11,22,13,26)(15,29,16,28)(17,32,21,38)(20,37,31,24)(23,39,30,25)(33,35,34,40);(9,13,14,11)(10,17,19,21)(12,26,27,22)(15,23,16,30)(18,32,36,38)(20,34,31,33)(24,35,37,40)(25,29,39,28)

Polycyclic presentation:

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

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, 302069);