#9224291

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	#47580	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,23,25)(19,20,27,21)(22,24,31,32)(26,28,30,29)(34,38)(36,43)(39,44)(42,47);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19)(18,20)(21,25)(22,30)(23,27)(24,29)(26,31)(28,32)(35,41)(39,44)(42,47)(46,48);(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,32,30)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,21,23,20)(18,19,25,27)(22,28,31,29)(24,30,32,26);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,22)(18,24)(19,26)(20,28)(21,29)(23,31)(25,32)(27,30)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,19)(18,21)(20,25)(22,30)(23,27)(24,28)(26,31)(29,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9131487);