#8534919

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

3946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=h,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, 8442115);