#8968300

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

4712 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)(3,4)(5,6)(7,8)(9,10,15,13)(11,17,12,16)(14,21,22,18)(19,23,20,24)(25,26,30,32)(27,31,35,39)(28,33,36,38)(29,34,37,40);(2,6)(4,7)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,27)(26,31)(28,29)(30,35)(32,39)(33,34)(36,37)(38,40);(1,2,5,6)(3,7,8,4)(10,13)(14,22)(16,17)(19,20)(26,32)(27,35)(29,37)(33,38);(1,3)(2,7)(4,6)(5,8)(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,28)(26,33)(27,29)(30,36)(31,34)(32,38)(35,37)(39,40);(1,4,5,7)(2,3,6,8)(9,13,15,10)(11,16,12,17)(14,21,22,18)(19,23,20,24)(25,29)(26,34)(27,36)(28,35)(30,37)(31,38)(32,40)(33,39);(1,3)(2,7)(4,6)(5,8)(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=hi,e^2=g,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, 8875496);