#510071

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,35)(28,37)(29,36)(30,40)(31,41)(32,39)(33,38)(34,42)(43,46)(44,47)(45,48);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(19,27)(20,30)(21,32)(23,35)(25,40)(26,39)(28,45)(29,44)(31,34)(33,43)(36,47)(37,48)(38,46)(41,42);(10,14)(13,16)(17,19,22,29)(18,23,24,36)(20,31,33,45)(21,34,32,28)(25,41,38,48)(26,42,39,37)(27,43,44,30)(35,46,47,40);(2,6)(3,8)(10,14)(13,16)(18,24)(23,36)(25,38)(26,39)(35,47)(37,42)(40,46)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417267);