#112165

order	512
nilpotency class	3
isoclinism	isoclinic class of #112162
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

831 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 831 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,9,11,15)(4,7)(5,8)(6,10,14,12)(13,16)(17,18,22,27,23,26,21,38)(19,28,32,43,30,24,31,40)(20,33,36,44,37,25,35,42)(29,45,47,48,46,39,34,41);(1,3)(2,6)(4,12)(5,11)(7,15)(8,14)(9,16)(10,13)(17,19)(18,24)(20,34)(21,32)(22,31)(23,30)(25,41)(26,28)(27,43)(29,35)(33,48)(36,46)(37,47)(38,40)(39,42)(44,45);(4,13)(6,14)(7,16)(10,12)(17,20,23,37)(18,25,26,33)(19,29,30,46)(21,36,22,35)(24,39,28,45)(27,44,38,42)(31,47,32,34)(40,48,43,41)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19361);