#508977

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

448 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,23)(19,27)(25,37)(26,38)(29,44)(30,35)(33,47)(41,45);(2,5)(6,8)(17,18,21,25,32,38,46,45,20,23,31,37,22,26,34,41)(19,28,29,42,35,48,47,36,27,43,44,39,30,40,33,24);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19,20,27)(18,24,23,36)(21,33,31,47)(22,35,32,30)(25,40,37,48)(26,42,38,39)(28,45,43,41)(29,46,44,34);(1,3,4,7)(2,6,5,8)(19,27)(24,36)(28,43)(29,44)(30,35)(33,47)(39,42)(40,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=f,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,g^c=gh,h^c=hi

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, 416173);