#9054484

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

4322 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961680);