#117306

order	512
nilpotency class	3
isoclinism	isoclinic class of #117305
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

541 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)(3,9,11,6)(4,8)(5,7)(10,16,13,14)(12,15)(17,18)(19,28)(20,32,36,44)(21,37)(22,26)(23,27)(24,30)(25,35,43,34)(29,42,33,39)(31,40)(38,41)(45,46,48,47);(3,10)(4,12)(6,14)(8,15)(9,16)(11,13)(17,19,21,30,23,31,22,38)(18,24,26,28,27,41,37,40)(20,33,34,47,36,29,35,46)(25,42,32,45,43,39,44,48);(1,3)(2,6)(4,13)(5,11)(7,9)(8,16)(10,12)(14,15)(17,20,23,36)(18,25,27,43)(19,29,31,33)(21,35,22,34)(24,39,41,42)(26,44,37,32)(28,45,40,48)(30,47,38,46)

Polycyclic presentation:

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

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, 24502);