#8471124

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

4597 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)(3,7)(4,8)(5,6)(9,10,15,18)(11,19,22,16)(12,23,21,14)(13,24,20,17)(25,26,32,30)(27,37,39,33)(28,35,29,34)(31,36,38,40);(2,6)(3,8)(9,11)(10,16)(12,20)(13,21)(14,17)(15,22)(18,19)(23,24)(25,27)(26,33)(28,38)(29,31)(30,37)(32,39)(34,40)(35,36);(1,3)(2,4)(5,8)(6,7)(25,28,32,29)(26,34,30,35)(27,38,39,31)(33,40,37,36);(1,3)(2,4)(5,8)(6,7)(9,12,15,21)(10,14,18,23)(11,20,22,13)(16,17,19,24)(25,29,32,28)(26,35,30,34)(27,38,39,31)(33,40,37,36);(1,4)(2,3)(5,7)(6,8)(9,13,15,20)(10,17,18,24)(11,21,22,12)(14,19,23,16)(25,30,32,26)(27,37,39,33)(28,35,29,34)(31,40,38,36);(9,14)(10,12)(11,17)(13,19)(15,23)(16,20)(18,21)(22,24)(25,31,32,38)(26,36,30,40)(27,29,39,28)(33,35,37,34)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=gi,e^2=gi,f^2=hi,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, 8378320);