#113180

order	512
nilpotency class	3
isoclinism	isoclinic class of #113169
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

832 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)(3,11,14,8)(4,7)(5,9)(6,10)(12,16,13,15)(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);(2,7)(3,12)(4,5)(9,10)(13,14)(15,16)(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);(1,3,6,14)(2,8,10,11)(4,12,5,13)(7,15,9,16)(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=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,f^c=fh,g^c=gh

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, 20376);