#9017191

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=i,e^2=ghi,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^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924387);