#116126

order	512
nilpotency class	3
isoclinism	isoclinic class of #116120
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

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,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, 23322);