#8932202

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

4309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=h,e^2=gi,f^2=h,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, 8839398);