#9018084

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

4068 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,f^2=i,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, 8925280);