#8818486

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

4129 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)(4,9,13,15)(5,10,14,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(34,39)(35,41)(37,45)(43,48);(2,7)(5,14)(8,11)(9,15)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,34,38,39)(35,36,41,44)(37,46,45,40)(42,48,47,43);(3,12)(4,13)(8,11)(9,15)(18,19)(20,27)(24,26)(28,32)(33,35,38,41)(34,36,39,44)(37,42,45,47)(40,43,46,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,16)(14,15)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,4)(2,9)(3,10)(5,8)(6,13)(7,15)(11,14)(12,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,5)(2,10)(3,9)(4,8)(6,14)(7,16)(11,13)(12,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725682);