#9206433

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

3944 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113629);