#8535199

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

4369 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)(18,21)(19,20)(25,29)(26,28)(34,39)(36,41)(40,46)(42,47);(3,4)(7,10)(11,12)(15,16)(17,18,23,21)(19,24,20,27)(22,25,30,29)(26,31,28,32)(33,34,38,39)(35,40,44,46)(36,37,41,45)(42,43,47,48);(1,3,6,4)(2,7,9,10)(5,12,14,11)(8,16,13,15)(17,19,23,20)(18,24,21,27)(22,26,30,28)(25,31,29,32)(33,35,38,44)(34,40,39,46)(36,42,41,47)(37,43,45,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,19)(18,24,21,27)(22,26,30,28)(25,32,29,31)(33,36)(34,37)(35,42)(38,41)(39,45)(40,43)(44,47)(46,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21,23,18)(19,24,20,27)(22,29,30,25)(26,31,28,32)(33,35)(34,40)(36,42)(37,43)(38,44)(39,46)(41,47)(45,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)(33,37)(34,41)(35,43)(36,39)(38,45)(40,47)(42,46)(44,48)

Polycyclic presentation:

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

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, 8442395);