#8434730

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

4550 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=g,e^2=h,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, 8341926);