#8630831

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

4182 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=gh,e^2=i,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, 8538027);