#8456929

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	#42988	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4519 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8364125);