#8843598

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

4254 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=ghi,e^2=gh,f^2=gh,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, 8750794);