#112282

order	512
nilpotency class	3
isoclinism	isoclinic class of #112268
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1196 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1196 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,g^b=ghi,d^c=dh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19478);