#8798587

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	#45741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4288 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gi,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8705783);