#512500

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

504 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 166 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)(17,18,23,28)(19,29,31,27)(20,32,36,46)(21,37,38,26)(22,24,39,41)(25,42,45,35)(30,44,33,48)(34,43,47,40);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,22,23,31,38,39)(18,24,26,29,28,41,37,27)(20,33,34,42,36,30,47,35)(25,43,44,32,45,40,48,46);(1,3,5,4)(2,7,6,8)(17,20,31,33,21,34,39,42,23,36,19,30,38,47,22,35)(18,25,24,40,26,44,29,46,28,45,41,43,37,48,27,32);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,19,21,22,23,31,38,39)(18,24,26,29,28,41,37,27)(20,30,34,35,36,33,47,42)(25,40,44,46,45,43,48,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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

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, 419696);