#8664854

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	#45732	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4475 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8572050);