#496080

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
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

438 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 242 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,20,32,33,23,27,35,41,36,34,25,40)(19,24,30,39,31,29,45,46,28,37,43,47,44,42,38,48);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,28)(25,38)(26,39)(27,37)(32,45)(33,46)(34,42)(35,43)(36,44)(40,48)(41,47);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(19,28)(24,37)(29,42)(30,43)(31,44)(38,45)(39,47)(46,48);(1,4,5,3)(2,8,6,7)(17,20,21,33,22,27,32,41,23,34,35,40,36,18,25,26)(19,29,30,46,31,37,45,47,28,42,43,48,44,24,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403276);