#8467200

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

4298 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=gh,e^2=ghi,f^2=gi,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, 8374396);