#8461734

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

4560 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)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26)(27,31)(28,33)(29,34)(30,35)(32,37)(36,38)(39,40);(2,6)(4,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27,30,37)(26,31,35,32)(28,34,38,40)(29,36,39,33);(1,3)(2,7)(4,6)(5,8)(25,27)(26,32)(28,34)(29,36)(30,37)(31,35)(33,39)(38,40);(1,4,5,7)(2,3,6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,39)(38,40);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,30,38)(26,33,35,36)(27,34,37,40)(29,32,39,31);(1,3)(2,4)(5,8)(6,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29)(26,34)(27,36)(28,32)(30,39)(31,38)(33,37)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,e^2=i,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, 8368930);