#9720338

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

4415 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=g,f^2=h,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, 9627534);