#8533296

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

4106 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)(3,8)(4,10)(5,6)(7,9)(11,13)(12,14)(15,16)(17,18,23,26)(19,24,29,20)(21,22,25,31)(27,28,32,30)(34,40)(36,46)(41,43)(44,48);(1,3,7,12)(2,8,9,14)(4,13,15,5)(6,10,11,16)(17,19,23,29)(18,24,26,20)(21,27,25,32)(22,28,31,30)(33,34,39,40)(35,41,38,43)(36,42,46,37)(44,45,48,47);(2,9)(3,12)(5,13)(10,16)(17,20,23,24)(18,19,26,29)(21,30,25,28)(22,27,31,32)(33,35,39,38)(34,41,40,43)(36,44,46,48)(37,47,42,45);(1,4)(2,10)(3,5)(6,8)(7,15)(9,16)(11,14)(12,13)(17,21)(18,22)(19,27)(20,30)(23,25)(24,28)(26,31)(29,32)(33,36)(34,42)(35,44)(37,40)(38,48)(39,46)(41,45)(43,47);(1,5)(2,11)(3,4)(6,9)(7,13)(8,16)(10,14)(12,15)(17,20)(18,19)(21,30)(22,27)(23,24)(25,28)(26,29)(31,32)(33,37,39,42)(34,36,40,46)(35,45,38,47)(41,48,43,44);(1,6,7,11)(2,5,9,13)(3,10,12,16)(4,14,15,8)(17,22)(18,25)(19,28)(20,27)(21,26)(23,31)(24,32)(29,30)(33,38,39,35)(34,43,40,41)(36,44,46,48)(37,47,42,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,e^2=g,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, 8440492);