#508975

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

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

normal subgroups only

This group has 456 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)(24,36)(25,37)(26,38)(27,43)(39,42)(40,47)(41,48);(2,5)(6,8)(17,18,21,25,32,38,46,48,20,23,31,37,22,26,34,41)(19,27,29,42,35,47,45,36,28,43,44,39,30,40,33,24);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,28)(21,33)(22,35)(23,36)(25,40)(26,42)(27,41)(29,34)(30,32)(31,45)(37,47)(38,39)(43,48)(44,46);(1,3,4,7)(2,6,5,8)(19,28)(24,36)(27,43)(29,44)(30,35)(33,45)(39,42)(40,47)

Polycyclic presentation:

2^9|a^2=f,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, 416171);