#8977350

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8884546);