#8186850

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

4146 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)(3,7)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(34,39)(35,36)(42,46)(43,44);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(18,24)(19,20)(26,29)(27,28)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,8)(3,9)(10,13)(11,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35,38,36)(34,40,39,41)(37,44,45,43)(42,48,46,47);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,37,38,45)(34,42,39,46)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094046);