#8096726

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

4673 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=h,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, 8003922);