#8471184

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

4597 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=i,e^2=hi,f^2=hi,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, 8378380);