#160411

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; only the 470 normal subgroups are shown.

normal subgroups only

This group has 1960 conjugacy classes of subgroups — too many to draw. Only its 470 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=efh,d^2=eg,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, 67607);