#8534934

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

3948 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,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, 8442130);