#8718494

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	#43913	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4487 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625690);