#9240938

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
prime radical	#51582	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9148134);