#499703

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 460 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)(17,18,22,24)(19,23,30,36)(20,25,33,39)(21,26,32,40)(27,35,44,46)(28,37,43,47)(29,38,34,42)(31,41,45,48);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,27)(21,34)(22,30)(24,36)(25,35)(26,42)(28,31)(29,32)(33,44)(37,41)(38,40)(39,46)(43,45)(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)(25,41)(26,40)(28,29)(30,44)(33,45)(34,43)(36,46)(37,38)(39,48)(42,47);(1,4,5,3)(2,6,8,7)(18,24)(23,36)(25,39)(26,40)(35,46)(37,47)(38,42)(41,48)

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,d^a=di,c^b=cg,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, 406899);