#8934291

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

4480 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841487);