#8936901

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	#40920	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4420 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8844097);