#8587478

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

4018 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=g,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, 8494674);