#8618008

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

4015 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8525204);