#8932979

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
prime radical	#51815	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4518 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,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, 8840175);