#8986895

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

4070 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,22,25)(19,23,26,20)(21,24,29,31)(27,30,32,28)(33,34,39,42)(35,43,45,40)(36,46,47,37)(38,41,44,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,35,39,45)(34,40,42,43)(36,41,47,48)(37,44,46,38);(2,6)(3,10)(4,13)(14,15)(19,26)(20,23)(21,29)(24,31)(36,47)(37,46)(38,44)(41,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20,22,23)(18,19,25,26)(21,30,29,28)(24,32,31,27)(33,36,39,47)(34,37,42,46)(35,41,45,48)(38,40,44,43);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,36)(35,44)(38,45)(39,46)(40,41)(42,47)(43,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,38,39,44)(34,41,42,48)(35,37,45,46)(36,40,47,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=ghi,f^2=gh,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, 8894091);