#8750034

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

4180 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8657230);