#8621703

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

4005 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)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(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);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(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);(2,5)(8,10)(9,12)(15,16)(18,25)(19,27)(21,32)(28,30)(35,40)(37,45)(43,46)(44,48);(1,4)(2,9)(3,11)(5,12)(6,13)(7,14)(8,15)(10,16)(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);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(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);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(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=ghi,b^2=hi,d^2=gh,e^2=hi,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, 8528899);