#8661376

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

4031 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,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,f^d=fgi,f^e=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, 8568572);