#397503

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 304699);