#9175000

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

4296 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)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,18,23,24)(19,27,28,20)(21,25,31,32)(22,30,29,26)(33,34,39,44)(35,38,43,40)(36,41,46,48)(37,42,45,47);(1,3,6,10)(2,7,9,4)(5,11,14,16)(8,15,13,12)(17,19,23,28)(18,20,24,27)(21,29,31,22)(25,30,32,26)(33,35)(34,40)(36,45)(37,46)(38,44)(39,43)(41,42)(47,48);(3,10)(4,7)(11,16)(12,15)(18,24)(20,27)(25,32)(26,30)(33,36,39,46)(34,41,44,48)(35,45,43,37)(38,47,40,42);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,20,23,27)(18,19,24,28)(21,30,31,26)(22,25,29,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,42)(44,48);(1,2)(3,7)(4,10)(5,13)(6,9)(8,14)(11,12)(15,16)(17,21,23,31)(18,25,24,32)(19,22,28,29)(20,26,27,30)(33,37,39,45)(34,42,44,47)(35,46,43,36)(38,48,40,41);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22)(18,26)(19,21)(20,25)(23,29)(24,30)(27,32)(28,31)(33,38)(34,43)(35,44)(36,47)(37,41)(39,40)(42,46)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=h,e^2=g,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, 9082196);