#8326001

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

4437 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8233197);