#8963719

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

4138 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8870915);