#8595722

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

4605 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)(3,8)(4,5)(6,7)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(1,3,6,5)(2,4,7,8)(9,11,14,18)(10,15,17,22)(12,19,16,23)(13,20,21,24)(26,31)(32,34)(33,38)(39,40);(2,7)(4,8)(11,18)(15,22)(19,23)(20,24)(25,27,30,28)(26,32,31,34)(29,35,37,36)(33,39,38,40);(2,7)(3,5)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,28,30,27)(26,32,31,34)(29,35,37,36)(33,40,38,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(1,5,6,3)(2,4,7,8)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29,30,37)(26,33,31,38)(27,35,28,36)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8502918);