#8843509

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

4190 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8750705);