#110188

order	512
nilpotency class	3
isoclinism	isoclinic class of #110180
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

607 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fhi

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, 17384);