#8581786

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

3950 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=gi,e^2=ghi,f^2=hi,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, 8488982);