#8806872

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

4058 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8714068);