#8922648

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

4114 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34,39,38)(35,44,47,40)(36,41,46,42)(37,43,45,48);(2,7)(4,9)(8,10)(12,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35)(34,40)(36,45)(37,46)(38,44)(39,47)(41,43)(42,48);(2,7)(3,11)(5,15)(12,14)(19,25)(20,28)(23,27)(24,29)(33,36,39,46)(34,41,38,42)(35,45,47,37)(40,43,44,48);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,36)(34,42)(35,45)(37,47)(38,41)(39,46)(40,48)(43,44);(1,4,6,9)(2,5,7,15)(3,12,11,14)(8,16,10,13)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,37,39,45)(34,43,38,48)(35,46,47,36)(40,41,44,42);(1,5)(2,9)(3,13)(4,7)(6,15)(8,12)(10,14)(11,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,38,39,34)(35,44,47,40)(36,41,46,42)(37,48,45,43)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=i,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829844);