#8534741

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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=ghi,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, 8441937);