#8828640

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

4114 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)(3,5,10,8)(4,12,13,9)(11,16,14,15)(17,18,22,25)(19,23,26,20)(21,24,30,32)(27,31,29,28)(33,34,38,41)(35,37,42,39)(36,40,46,48)(43,45,44,47);(2,7)(4,13)(5,8)(11,14)(17,19,22,26)(18,23,25,20)(21,29,30,27)(24,28,32,31)(35,42)(37,39)(43,44)(45,47);(2,7)(3,10)(9,12)(11,14)(19,26)(20,23)(27,29)(28,31)(33,35)(34,39)(36,44)(37,41)(38,42)(40,45)(43,46)(47,48);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,29)(30,32)(33,35)(34,39)(36,44)(37,41)(38,42)(40,45)(43,46)(47,48);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,37,38,39)(34,35,41,42)(36,45,46,47)(40,43,48,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,e^2=g,f^2=h,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, 8735836);