#8533286

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

4127 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)(3,8)(4,10)(5,6)(7,9)(11,13)(12,14)(15,16)(18,24)(19,20)(21,29)(30,32)(33,34,39,42)(35,40,45,36)(37,38,41,47)(43,44,48,46);(1,3,7,12)(2,8,9,14)(4,13,15,5)(6,10,11,16)(17,18,23,24)(19,22,20,27)(21,25,29,31)(26,30,28,32)(33,35,39,45)(34,40,42,36)(37,43,41,48)(38,44,47,46);(2,9)(3,12)(5,13)(10,16)(17,19,23,20)(18,22,24,27)(21,28,29,26)(25,32,31,30)(33,36,39,40)(34,35,42,45)(37,46,41,44)(38,43,47,48);(1,4)(2,10)(3,5)(6,8)(7,15)(9,16)(11,14)(12,13)(17,20,23,19)(18,22,24,27)(21,28,29,26)(25,30,31,32)(33,37)(34,38)(35,43)(36,46)(39,41)(40,44)(42,47)(45,48);(1,5)(2,11)(3,4)(6,9)(7,13)(8,16)(10,14)(12,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,36)(34,35)(37,46)(38,43)(39,40)(41,44)(42,45)(47,48);(1,6,7,11)(2,5,9,13)(3,10,12,16)(4,14,15,8)(17,22)(18,20)(19,24)(21,30)(23,27)(25,28)(26,31)(29,32)(33,38)(34,41)(35,44)(36,43)(37,42)(39,47)(40,48)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^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, 8440482);