#8968135

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

4648 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)(3,7)(4,8)(5,6)(9,10,15,13)(11,16,18,12)(14,21,22,17)(19,20,24,23)(25,26)(27,33)(28,32)(29,37)(30,31)(34,38)(35,39)(36,40);(2,6)(3,8)(9,11,15,18)(10,16,13,12)(14,19,22,24)(17,23,21,20)(26,31)(32,33)(34,37)(39,40);(1,3)(2,4)(5,8)(6,7)(10,13)(11,18)(14,22)(20,23)(25,27,30,28)(26,32,31,33)(29,36,38,35)(34,40,37,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,18)(14,20)(15,16)(17,19)(21,24)(22,23)(25,28,30,27)(26,33,31,32)(29,36,38,35)(34,40,37,39);(1,2)(3,4)(5,6)(7,8)(9,13,15,10)(11,16,18,12)(14,21,22,17)(19,23,24,20)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,39)(37,38);(1,4)(2,3)(5,7)(6,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,30,38)(26,34,31,37)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=hi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8875331);