#9214252

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

4171 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 9121448);