#8388396

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

4430 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8295592);