#8201970

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

4095 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109166);