#8743431

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	#37833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

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=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8650627);