#8821815

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

4052 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=h,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8729011);