#9232845

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

4222 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)(4,6)(5,12)(7,11)(9,14)(10,15)(13,16)(17,18,23,27)(19,24,29,22)(20,25,30,31)(21,26,28,32)(34,40)(36,44)(41,43)(45,47);(1,2,6,4)(3,7,11,8)(5,9,14,12)(10,15,13,16)(17,19)(18,24)(20,21)(22,27)(23,29)(25,26)(28,30)(31,32)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,48)(46,47);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,16)(13,14)(17,20)(18,25)(19,28)(21,29)(22,26)(23,30)(24,32)(27,31)(33,35,39,37)(34,41,40,43)(36,45,44,47)(38,46,42,48);(1,4,6,2)(3,8,11,7)(5,9,14,12)(10,15,13,16)(17,21,23,28)(18,26,27,32)(19,20,29,30)(22,31,24,25)(33,36)(34,42)(35,45)(37,47)(38,40)(39,44)(41,48)(43,46);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,20)(18,25)(19,21)(22,32)(23,30)(24,26)(27,31)(28,29)(33,37,39,35)(34,43,40,41)(36,45,44,47)(38,46,42,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,22,23,24)(18,19,27,29)(20,26,30,32)(21,25,28,31)(33,38,39,42)(34,44,40,36)(35,46,37,48)(41,47,43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=hi,e^2=gh,f^2=ghi,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, 9140041);