#508710

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 492 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)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(17,18,20,24)(19,27,29,23)(21,25,31,38)(22,26,32,39)(28,43,45,36)(30,42,35,37)(33,40,44,48)(34,47,46,41);(9,11,13,16)(10,14,12,15)(19,28)(21,33)(22,32)(23,36)(25,40)(26,39)(27,43)(29,45)(30,46)(31,44)(34,35)(37,47)(38,48)(41,42);(2,5)(6,8)(10,12)(14,15)(17,19,20,29)(18,23,24,27)(21,34,31,46)(22,35,32,30)(25,41,38,47)(26,42,39,37)(28,44,45,33)(36,48,43,40);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(18,24)(23,27)(25,38)(26,39)(36,43)(37,42)(40,48)(41,47)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,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, 415906);