#112853

order	512
nilpotency class	3
isoclinism	isoclinic class of #112845
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1134 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1134 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:

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

Polycyclic presentation:

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

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, 20049);