#508972

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

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

normal subgroups only

This group has 480 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)(17,18,21,26)(19,28,32,24)(20,35,23,25)(22,27,29,38)(30,43,47,39)(31,40,34,44)(33,45,46,41)(36,48,37,42);(2,5)(6,8)(17,19,29,46,20,31,37,30,21,32,22,33,23,34,36,47)(18,24,38,45,35,44,42,39,26,28,27,41,25,40,48,43);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(19,30)(20,23)(22,37)(24,39)(25,35)(27,42)(28,43)(29,36)(31,46)(32,47)(33,34)(38,48)(40,41)(44,45);(3,7)(6,8)(17,20,21,23)(18,25,26,35)(19,31,32,34)(22,36,29,37)(24,40,28,44)(27,42,38,48)(30,33,47,46)(39,45,43,41)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,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, 416168);