#8990313

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

4252 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(33,34,39,37)(35,40,38,45)(36,41,43,46)(42,47,44,48);(2,6)(7,9)(8,12)(14,15)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35,39,38)(34,40,37,45)(36,42,43,44)(41,47,46,48);(2,6)(3,10)(4,13)(14,15)(18,19)(20,27)(21,29)(30,31)(34,37)(35,38)(36,43)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36)(34,41)(35,42)(37,46)(38,44)(39,43)(40,47)(45,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37,39,34)(35,40,38,45)(36,41,43,46)(42,48,44,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,35)(34,40,37,45)(36,44,43,42)(41,47,46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8897509);