#8185333

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

4473 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092529);