#160407

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

1960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=egi,b^2=efg,c^2=egh,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67603);