#8258122

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

4092 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165318);