#8814035

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

4568 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721231);