#8096134

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

4510 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003330);