#8340038

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

4147 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)(3,7)(4,11)(5,10)(6,16)(8,13)(9,15)(12,14)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(33,34,39,42)(35,41,37,40)(36,45,46,38)(43,44,47,48);(1,3)(2,9)(4,14)(5,6)(7,13)(8,15)(10,12)(11,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,39,37)(34,40,42,41)(36,43,46,47)(38,44,45,48);(1,4,8,5)(2,10,13,11)(3,14,15,6)(7,16,9,12)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35,39,37)(34,41,42,40)(36,43,46,47)(38,48,45,44);(1,5,8,4)(2,11,13,10)(3,14,15,6)(7,16,9,12)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,36)(34,38)(35,43)(37,47)(39,46)(40,44)(41,48)(42,45);(1,6,8,14)(2,12,13,16)(3,5,15,4)(7,11,9,10)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,39,35)(34,40,42,41)(36,43,46,47)(38,48,45,44);(1,7,8,9)(2,3,13,15)(4,12,5,16)(6,11,14,10)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,47)(42,46)

Polycyclic presentation:

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

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, 8247234);