#8932230

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

4569 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839426);