#9715469

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

4528 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:

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

Polycyclic presentation:

2^9|a^2=gh,e^2=i,f^2=gi,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, 9622665);