#9625397

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

4225 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532593);