#8456959

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4505 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=gh,e^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, 8364155);