#496071

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator, socle	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

486 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 234 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,27,34,40,23,20,25,32,35,33,39,45)(19,24,30,37,31,38,44,47,28,29,36,42,41,43,46,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,28)(25,36)(26,37)(27,38)(32,42)(33,43)(34,44)(35,41)(39,46)(40,47)(45,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,28)(24,29)(30,36)(31,41)(37,42)(38,43)(44,46)(47,48);(17,20,21,32,22,33,34,45,23,18,25,26,35,27,39,40)(19,29,30,42,31,43,44,48,28,24,36,37,41,38,46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403267);