#8691865

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

4147 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,21,24)(19,25,27,22)(20,23,30,28)(26,32,29,31)(33,34,38,39)(35,40,36,41)(37,42,46,45)(43,47,44,48);(2,7)(3,10)(9,12)(11,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32)(34,39)(40,41)(42,45)(47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31)(33,36,38,35)(34,41,39,40)(37,43,46,44)(42,47,45,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8599061);