#499700

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

468 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 170 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)(9,10,13,16)(11,15,12,14)(17,18,22,26)(19,23,28,36)(20,24,33,41)(21,25,32,40)(27,35,43,46)(29,37,44,47)(30,38,34,42)(31,39,45,48);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,27)(21,34)(22,28)(24,35)(25,42)(26,36)(29,31)(30,32)(33,43)(37,39)(38,40)(41,46)(44,45)(47,48);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,15,16,14)(19,27)(20,31)(21,32)(23,35)(24,39)(25,40)(28,43)(29,30)(33,45)(34,44)(36,46)(37,38)(41,48)(42,47);(1,4,5,3)(2,6,8,7)(19,28)(23,36)(27,43)(29,44)(30,34)(35,46)(37,47)(38,42)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=e,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406896);