#8718457

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	#38367	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 8625653);