#8533597

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

4107 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)(3,4)(5,12)(6,9)(7,8)(10,11)(13,14)(15,16)(18,21)(19,20)(25,29)(26,28)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(1,3,7,5)(2,4,8,12)(6,10,15,13)(9,11,16,14)(17,18,23,21)(19,24,20,27)(22,25,30,29)(26,31,28,32)(36,44)(41,46)(43,45)(47,48);(2,8)(3,5)(9,16)(10,13)(17,19,23,20)(18,24,21,27)(22,26,30,28)(25,31,29,32)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(1,4)(2,3)(5,8)(6,14)(7,12)(9,13)(10,16)(11,15)(17,20,23,19)(18,24,21,27)(22,26,30,28)(25,32,29,31)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5,7,3)(2,4,8,12)(6,13,15,10)(9,11,16,14)(17,21,23,18)(19,24,20,27)(22,29,30,25)(26,31,28,32)(33,37,39,35)(34,42,38,40)(36,45,44,43)(41,48,46,47);(1,6,7,15)(2,9,8,16)(3,10,5,13)(4,11,12,14)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=g,e^2=h,f^2=gh,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, 8440793);