#8464340

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4217 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,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, 8371536);