#8622048

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

4034 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)(3,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,27)(20,29)(21,28)(22,26)(23,24)(25,31)(30,32)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(1,2)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,19,23,28)(18,21,24,27)(20,30,31,22)(25,32,29,26)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,4)(7,9)(8,12)(15,16)(18,24)(21,27)(25,29)(26,32)(35,40)(37,45)(43,46)(44,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20,23,31)(18,25,24,29)(19,22,28,30)(21,26,27,32)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21)(18,19)(20,26)(22,25)(23,27)(24,28)(29,30)(31,32)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22)(18,26)(19,20)(21,25)(23,30)(24,32)(27,29)(28,31)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529244);