#8534855

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

4315 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)(3,8)(4,7)(5,9)(6,10)(11,15)(12,14)(13,16)(18,24)(19,20)(21,29)(30,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,18)(19,25)(20,22)(21,26)(23,24)(27,32)(28,30)(29,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,3,6,4)(2,8,10,7)(5,11,13,12)(9,15,16,14)(17,19,23,20)(18,25,24,22)(21,28,29,27)(26,30,31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4,6,3)(2,8,10,7)(5,11,13,12)(9,14,16,15)(17,20,23,19)(18,25,24,22)(21,28,29,27)(26,32,31,30)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,21,23,29)(18,26,24,31)(19,27,20,28)(22,30,25,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(17,22,23,25)(18,20,24,19)(21,30,29,32)(26,28,31,27)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=g,e^2=hi,f^2=g,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, 8442051);