#498781

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

426 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cefi,d^b=de,d^c=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, 405977);