#375716

order	512
nilpotency class	3
isoclinism	isoclinic class of #196002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1130 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 258 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)(17,18,21,26,22,25,30,39)(19,23,29,36,28,35,41,47)(20,24,32,37,31,38,33,40)(27,34,43,46,42,45,44,48);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,34)(25,35)(26,36)(30,41)(31,42)(32,43)(33,44)(37,46)(38,45)(39,47)(40,48);(10,13)(14,16)(17,20,30,32,22,31,21,33)(18,24,26,40,25,38,39,37)(19,27,41,43,28,42,29,44)(23,34,36,48,35,45,47,46);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,28)(23,35)(27,42)(29,41)(34,45)(36,47)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fhi,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,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, 282912);