#8534543

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

3939 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:

(1,2,8,11)(3,12,7,9)(4,6,5,10)(13,15,14,16)(18,24)(20,30)(25,27)(28,32);(1,3,8,7)(2,9,11,12)(4,14,5,13)(6,15,10,16)(17,18,23,24)(19,25,22,27)(20,26,30,21)(28,29,32,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(1,4,8,5)(2,10,11,6)(3,13,7,14)(9,15,12,16)(17,19,23,22)(18,25,24,27)(20,28,30,32)(21,31,26,29)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,5,8,4)(2,6,11,10)(3,14,7,13)(9,16,12,15)(17,20)(18,26)(19,28)(21,24)(22,32)(23,30)(25,29)(27,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,6)(2,4)(3,15)(5,11)(7,16)(8,10)(9,14)(12,13)(17,21,23,26)(18,20,24,30)(19,29,22,31)(25,32,27,28)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,7,8,3)(2,9,11,12)(4,14,5,13)(6,16,10,15)(17,22,23,19)(18,27,24,25)(20,28,30,32)(21,31,26,29)(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=g,c^2=ghi,d^2=h,e^2=gi,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, 8441739);