#8843832

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

4238 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)(17,18,23,26)(19,24,20,25)(21,31,32,22)(27,28,29,30)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(2,7)(4,10)(5,14)(11,15)(17,19,23,20)(18,24,26,25)(21,27,32,29)(22,30,31,28)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(2,7)(3,8)(9,13)(11,15)(19,20)(24,25)(27,29)(28,30)(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)(17,20,23,19)(18,25,26,24)(21,27,32,29)(22,30,31,28)(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)(18,22)(19,27)(20,29)(23,32)(24,30)(25,28)(26,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,22)(18,21)(19,28)(20,30)(23,31)(24,29)(25,27)(26,32)(33,37,38,46)(34,40,41,48)(35,43,39,45)(36,44,42,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=i,f^2=g,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, 8751028);