#8463884

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4431 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)(3,4,8,7)(9,10,16,15)(11,19,14,17)(12,18,13,22)(20,24,21,23)(26,29)(27,35)(28,38)(34,40);(2,6)(4,7)(9,11,16,14)(10,17,15,19)(12,20,13,21)(18,23,22,24)(25,26,31,29)(27,36,35,32)(28,39,38,33)(30,34,37,40);(1,3)(2,7)(4,6)(5,8)(9,12,16,13)(10,18,15,22)(11,20,14,21)(17,23,19,24)(25,27)(26,32)(28,37)(29,36)(30,38)(31,35)(33,40)(34,39);(1,4,5,7)(2,3,6,8)(9,13,16,12)(10,18,15,22)(11,21,14,20)(17,23,19,24)(25,28,31,38)(26,33,29,39)(27,37,35,30)(32,40,36,34);(9,14,16,11)(10,19,15,17)(12,21,13,20)(18,24,22,23)(25,29,31,26)(27,36,35,32)(28,39,38,33)(30,40,37,34);(1,3)(2,4)(5,8)(6,7)(9,15,16,10)(11,19,14,17)(12,22,13,18)(20,24,21,23)(25,30,31,37)(26,34,29,40)(27,38,35,28)(32,39,36,33)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=g,e^2=ghi,f^2=ghi,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, 8371080);