#8976390

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

4270 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)(3,8)(9,14)(10,15)(33,34)(35,41)(36,44)(37,40)(38,43)(39,46)(42,47)(45,48);(1,2,6,7)(3,4,8,11)(5,9,13,14)(10,12,15,16)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(3,8)(4,11)(5,13)(9,14)(18,23)(19,27)(20,30)(21,31)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,15)(14,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,35)(34,38)(36,42)(37,43)(39,47)(40,41)(44,48)(45,46);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,12,11,16)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,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, 8883586);