#8934594

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	#49749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,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, 8841790);