#508969

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 488 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,28)(20,34)(21,26)(22,27)(23,25)(24,31)(29,43)(30,39)(32,45)(33,44)(35,48)(36,42)(37,41)(38,46)(40,47);(2,5)(6,8)(17,19,22,32,20,30,35,46,21,31,36,47,23,33,37,29)(18,24,27,40,34,44,48,43,26,28,42,45,25,39,41,38);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(19,29)(20,23)(22,37)(24,38)(25,34)(27,41)(28,43)(30,47)(31,46)(32,33)(35,36)(39,40)(42,48)(44,45);(3,7)(6,8)(17,20,21,23)(18,25,26,34)(19,30,31,33)(22,35,36,37)(24,39,28,44)(27,41,42,48)(29,32,46,47)(38,45,43,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416165);