#8271132

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

4560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8178328);