#8445005

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

4549 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352201);