#8528880

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

4020 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)(3,11)(4,14)(5,10)(6,8)(7,13)(9,15)(12,16)(17,18)(19,24)(20,27)(21,25)(22,23)(26,32)(28,29)(30,31)(33,34,38,41)(35,39,44,47)(36,37,40,45)(42,43,46,48);(1,3,6,13)(2,7,8,11)(4,12,15,10)(5,9,16,14)(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,32)(30,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(2,8)(4,15)(7,11)(10,12)(18,23)(20,24)(25,31)(28,32)(33,35,38,44)(34,39,41,47)(36,42,40,46)(37,43,45,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19)(18,24)(20,23)(21,29)(22,27)(25,28)(26,30)(31,32)(33,36)(34,37)(35,42)(38,40)(39,43)(41,45)(44,46)(47,48);(1,4)(2,9)(3,12)(5,11)(6,15)(7,16)(8,14)(10,13)(17,20,22,24)(18,19,23,27)(21,28,30,32)(25,26,31,29)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5,6,16)(2,10,8,12)(3,9,13,14)(4,11,15,7)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,37)(34,40)(35,43)(36,41)(38,45)(39,46)(42,47)(44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,e^2=gh,f^2=g,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, 8436076);