#8873792

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

4096 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8780988);