#8324077

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

4539 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=h,e^2=gh,f^2=gi,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, 8231273);