#9059033

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

4122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8966229);