#9712827

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

4270 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)(3,8)(4,6)(5,7)(12,20)(15,22)(19,24)(21,23)(25,26,32,37)(27,33,39,30)(28,36,31,34)(29,40,38,35);(1,3,5,6)(2,4,7,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,27)(26,33)(28,29)(30,37)(31,38)(32,39)(34,35)(36,40);(9,11)(10,13)(12,19)(14,17)(15,21)(16,18)(20,24)(22,23)(25,28,32,31)(26,34,37,36)(27,38,39,29)(30,35,33,40);(9,12)(10,15)(11,19)(13,21)(14,20)(16,23)(17,24)(18,22)(25,29,32,38)(26,35,37,40)(27,28,39,31)(30,36,33,34);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,19)(13,18)(14,17)(15,23)(20,24)(21,22)(25,30)(26,27)(28,35)(29,36)(31,40)(32,33)(34,38)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13,14,16)(10,17,18,11)(12,21,20,23)(15,24,22,19)(25,31,32,28)(26,36,37,34)(27,38,39,29)(30,35,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9620023);