#8739227

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

4085 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8646423);