#512507

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 162 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,32)(20,36)(21,30)(22,25)(23,28)(24,31)(26,39)(27,34)(29,35)(33,44)(37,45)(38,43)(40,42)(41,47)(46,48);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,22,24,35,23,34)(18,25,28,32,31,27,30,29)(20,37,38,48,41,33,40,39)(26,43,44,36,46,42,45,47);(1,3,5,4)(2,7,6,8)(17,20,35,37,21,38,34,48,24,41,19,33,23,40,22,39)(18,26,25,42,28,44,32,47,31,46,27,43,30,45,29,36);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,19,21,22,24,35,23,34)(18,27,28,29,31,25,30,32)(20,37,38,48,41,33,40,39)(26,42,44,47,46,43,45,36)

Polycyclic presentation:

2^9|b^2=gi,c^2=ef,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^c=di

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, 419703);