#401340

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1299 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=ef,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,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308536);