#497168

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, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

408 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 226 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cei,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, 404364);