#116132

order	512
nilpotency class	3
isoclinism	isoclinic class of #116129
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

565 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fi,d^c=dh,f^c=fhi,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, 23328);