#8821792

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

4145 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)(19,23)(20,25)(21,30)(22,24)(26,31)(27,28)(29,32)(33,34,38,42)(35,36,40,39)(37,41,46,48)(43,44,45,47);(1,2,6,4)(3,7,9,10)(5,12,13,8)(11,16,15,14)(17,19)(18,23)(20,21)(22,27)(24,28)(25,30)(26,32)(29,31)(33,35,38,40)(34,39,42,36)(37,45,46,43)(41,44,48,47);(2,4)(3,9)(8,12)(11,15)(18,24)(19,27)(20,29)(26,30)(33,34,38,42)(35,39,40,36)(37,41,46,48)(43,47,45,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,22,29)(18,25,24,32)(19,21,27,31)(23,30,28,26)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,45,48,43);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21)(18,26)(19,29)(20,27)(22,31)(23,25)(24,30)(28,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,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, 8728988);