#8926441

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

4345 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)(3,5)(4,7)(6,8)(11,17)(16,23)(18,21)(19,24)(25,26,32,37)(27,38,40,33)(28,34,30,36)(29,35,39,31);(1,2,6,8)(3,5,7,4)(9,10,14,15)(11,16,17,23)(12,20,22,13)(18,19,21,24)(25,27,32,40)(26,33,37,38)(28,39,30,29)(31,34,35,36);(1,3,6,7)(2,4,8,5)(10,15)(13,20)(16,23)(19,24)(25,28,32,30)(26,34,37,36)(27,39,40,29)(31,33,35,38);(1,4,6,5)(2,3,8,7)(9,11)(10,16)(12,21)(13,19)(14,17)(15,23)(18,22)(20,24)(25,29)(26,35)(27,28)(30,40)(31,37)(32,39)(33,36)(34,38);(1,2)(3,4)(5,7)(6,8)(9,12)(10,13)(11,18)(14,22)(15,20)(16,24)(17,21)(19,23)(25,30,32,28)(26,36,37,34)(27,39,40,29)(31,33,35,38);(1,5,6,4)(2,7,8,3)(9,13)(10,12)(11,19)(14,20)(15,22)(16,21)(17,24)(18,23)(25,31,32,35)(26,29,37,39)(27,34,40,36)(28,38,30,33)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833637);