#8924714

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

4121 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=hi,e^2=i,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, 8831910);