#8534485

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

4123 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)(4,10)(5,15)(6,11)(7,9)(12,14)(13,16)(17,18)(19,29)(20,26)(21,23)(22,27)(24,28)(25,32)(30,31);(1,3)(2,8)(4,9)(5,12)(6,13)(7,10)(11,16)(14,15)(17,19)(18,25)(20,30)(21,27)(22,26)(23,31)(24,32)(28,29)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(2,9)(3,10)(5,11)(13,14)(17,20,24,21)(18,26,28,23)(19,30,32,27)(22,29,31,25)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,4,7,8)(2,10,9,3)(5,14,11,13)(6,16,15,12)(17,21,24,20)(18,26,28,23)(19,27,32,30)(22,29,31,25)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,5,7,11)(2,6,9,15)(3,12,10,16)(4,14,8,13)(17,22,24,31)(18,27,28,30)(19,26,32,23)(20,25,21,29)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,6)(2,11)(3,13)(4,12)(5,9)(7,15)(8,16)(10,14)(17,23,24,26)(18,21,28,20)(19,31,32,22)(25,27,29,30)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)

Polycyclic presentation:

2^9|c^2=ghi,d^2=gh,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, 8441681);