#8463915

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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)(3,4)(5,7)(6,8)(9,10)(11,19)(12,17)(13,14)(15,18)(16,23)(20,22)(21,24)(25,26,31,30)(27,32,38,37)(28,34,29,33)(35,40,36,39);(3,8)(4,6)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,27,31,38)(26,32,30,37)(28,35,29,36)(33,39,34,40);(1,3,5,8)(2,6,7,4)(9,12,15,13)(10,17,18,14)(11,20,23,21)(16,24,19,22)(25,28,31,29)(26,33,30,34)(27,35,38,36)(32,39,37,40);(1,4)(2,3)(5,6)(7,8)(9,13,15,12)(10,17,18,14)(11,21,23,20)(16,24,19,22)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,39)(34,40);(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,29,31,28)(26,34,30,33)(27,36,38,35)(32,40,37,39);(1,2)(3,6)(4,8)(5,7)(9,14,15,17)(10,13,18,12)(11,22,23,24)(16,21,19,20)(25,30,31,26)(27,37,38,32)(28,34,29,33)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=g,e^2=hi,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, 8371111);