#8968293

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=hi,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, 8875489);