#116196

order	512
nilpotency class	3
isoclinism	isoclinic class of #116195
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	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

570 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)(3,9,12,6)(4,13)(5,7)(8,15)(10,14,16,11)(17,18,22,27,23,28,21,26)(19,24,30,39,32,41,31,40)(20,33,37,25,35,44,36,43)(29,45,47,48,46,42,34,38);(1,2)(3,10)(4,13)(5,7)(6,11)(8,15)(9,14)(12,16)(17,19)(18,24)(20,34,35,47)(21,31)(22,30)(23,32)(25,42,43,45)(26,40)(27,39)(28,41)(29,36,46,37)(33,48,44,38);(1,3)(2,6)(4,14)(5,12)(7,9)(8,16)(10,13)(11,15)(17,20)(18,25)(19,29)(21,37)(22,36)(23,35)(24,38)(26,44)(27,33)(28,43)(30,34)(31,47)(32,46)(39,42)(40,45)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ehi,f^b=fi,d^c=dh,e^c=eh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23392);