#8630924

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

4183 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18,21,24)(19,22,26,25)(20,23,29,28)(27,30,32,31)(35,41)(38,42)(43,47)(46,48);(2,4)(3,10)(8,12)(11,13)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,34,37,40)(35,42,41,38)(36,39,45,44)(43,48,47,46);(2,4)(7,9)(8,12)(15,16)(19,26)(22,25)(27,32)(30,31)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,3)(2,7)(4,9)(5,13)(6,10)(8,16)(11,14)(12,15)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,31)(30,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,35)(34,38)(36,43)(37,41)(39,46)(40,42)(44,48)(45,47);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538120);