#135190

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1817
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^b=fi,e^c=ei,f^c=fg

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, 42386);