#8551283

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

4235 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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(18,19)(23,25)(24,26)(30,31)(33,34,38,40)(35,39,42,36)(37,41,46,47)(43,45,48,44);(2,7)(3,9)(10,15)(11,13)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35)(34,39)(36,40)(37,43)(38,42)(41,45)(44,47)(46,48);(2,7)(4,8)(10,15)(12,16)(20,27)(23,25)(28,32)(30,31)(34,40)(35,42)(41,47)(43,48);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36)(34,35)(37,45)(38,39)(40,42)(41,48)(43,47)(44,46);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,5,6,14)(2,10,7,15)(3,11,9,13)(4,12,8,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,e^2=g,f^2=hi,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, 8458479);