#158814

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

2112 conjugacy classes of subgroups; only the 498 normal subgroups are shown.

normal subgroups only

This group has 2112 conjugacy classes of subgroups — too many to draw. Only its 498 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=fi,d^2=fgh,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 66010);