#8934502

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4522 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)(9,10,14,15)(11,16,17,12)(13,21,22,18)(19,20,24,23)(25,26,30,33)(27,34,37,31)(28,38,32,29)(35,40,39,36);(1,3,5,8)(2,6,7,4)(10,15)(11,17)(18,21)(19,24)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,36,39,40);(1,3)(2,6)(4,7)(5,8)(10,15)(11,17)(13,22)(20,23)(25,27,30,37)(26,31,33,34)(28,35,32,39)(29,40,38,36);(2,7)(3,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,27)(26,31)(28,39)(29,36)(30,37)(32,35)(33,34)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,30,32)(26,29,33,38)(27,35,37,39)(31,40,34,36);(1,4)(2,3)(5,6)(7,8)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,29)(26,32)(27,36)(28,33)(30,38)(31,35)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,e^2=h,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, 8841698);