#8247667

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

4069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=gi,e^2=g,f^2=ghi,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, 8154863);