#8992972

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

4198 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8900168);