#9160756

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
prime radical	#50737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,15)(13,16)(17,18)(19,20)(21,30)(22,25)(23,31)(24,28)(26,29)(27,32)(33,34,40,37)(35,41,45,39)(36,38,42,47)(43,48,46,44);(1,3,7,12)(2,8,9,15)(4,13,14,5)(6,11,10,16)(17,19,24,25)(18,22,28,20)(21,23,29,32)(26,31,30,27)(33,35,40,45)(34,41,37,39)(36,46,42,43)(38,44,47,48);(2,9)(3,12)(4,14)(11,16)(17,20,24,22)(18,25,28,19)(21,31,29,27)(23,30,32,26)(33,36,40,42)(34,38,37,47)(35,43,45,46)(39,44,41,48);(1,4)(2,10)(3,13)(5,12)(6,9)(7,14)(8,16)(11,15)(17,21,24,29)(18,26,28,30)(19,23,25,32)(20,27,22,31)(33,37,40,34)(35,39,45,41)(36,38,42,47)(43,48,46,44);(1,5,7,13)(2,11,9,16)(3,14,12,4)(6,15,10,8)(17,22,24,20)(18,25,28,19)(21,27,29,31)(23,30,32,26)(33,38)(34,42)(35,44)(36,37)(39,46)(40,47)(41,43)(45,48);(1,6,7,10)(2,4,9,14)(3,11,12,16)(5,8,13,15)(17,23)(18,27)(19,29)(20,30)(21,25)(22,26)(24,32)(28,31)(33,39)(34,35)(36,44)(37,45)(38,43)(40,41)(42,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067952);