#112854

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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 816 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) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,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, 20050);