#113827

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

834 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=hi,f^2=h,g^2=hi,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, 21023);