#8324614

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

4470 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)(3,4)(9,10,15,18)(11,14,17,23)(12,13,16,22)(19,20,21,24)(25,26)(27,35)(28,32)(29,37)(30,33)(31,34)(36,40)(38,39);(1,2,6,5)(3,8,4,7)(9,11)(10,14)(12,19)(13,20)(15,17)(16,21)(18,23)(22,24)(25,27)(26,29)(28,36)(30,32)(31,37)(33,39)(34,35)(38,40);(1,3,6,4)(2,7,5,8)(11,17)(14,23)(19,21)(20,24)(25,28,31,39)(26,32,34,38)(27,36,37,33)(29,30,35,40);(1,4,6,3)(2,8,5,7)(9,12)(10,13)(11,19)(14,20)(15,16)(17,21)(18,22)(23,24)(25,29)(26,27)(28,30)(31,35)(32,36)(33,38)(34,37)(39,40);(1,5,6,2)(3,8,4,7)(9,13)(10,16)(11,20)(12,18)(14,21)(15,22)(17,24)(19,23)(25,30)(26,33)(27,32)(28,35)(29,39)(31,40)(34,36)(37,38);(9,14,15,23)(10,17,18,11)(12,20,16,24)(13,21,22,19)(25,26)(27,29)(28,38)(30,33)(31,34)(32,39)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231810);