#8621712

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

4018 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528908);