#8393372

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

4133 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:

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

Polycyclic presentation:

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

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, 8300568);