#8533565

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

4137 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440761);