#113823

order	512
nilpotency class	3
isoclinism	isoclinic class of #113815
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

811 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 21019);