#507513

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
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

474 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 178 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=egi,c^2=ef,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414709);