#8211668

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
prime radical	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8118864);