#8202377

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

4302 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18)(19,26)(20,25)(21,22)(23,27)(24,30)(28,29)(31,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(3,4)(5,14)(7,10)(8,13)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,32)(26,27)(35,43)(36,46)(39,48)(40,45);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,20,23,22)(18,25,27,21)(19,28,30,31)(24,32,26,29)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,48)(45,47);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21)(18,20)(19,29)(22,27)(23,25)(24,28)(26,31)(30,32);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22,23,20)(18,21,27,25)(19,28,30,31)(24,32,26,29)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,19)(18,26)(20,31)(21,29)(22,28)(23,30)(24,27)(25,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=h,e^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, 8109573);