#8247609

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

4181 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)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(3,4)(5,14)(7,10)(8,13)(17,18)(19,22)(20,24)(21,29)(23,26)(25,31)(27,32)(28,30)(34,38)(39,45)(40,43)(46,47);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,20)(18,24,26,22)(21,27,31,28)(25,32,29,30)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,19)(18,24,26,22)(21,28,31,27)(25,32,29,30)(33,36)(34,40)(35,41)(37,44)(38,43)(39,46)(42,48)(45,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,22,23,24)(18,19,26,20)(21,30,31,32)(25,27,29,28)(33,35)(34,39)(36,41)(37,42)(38,45)(40,46)(43,47)(44,48)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,f^2=hi,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, 8154805);