#8467212

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8374408);