#8700375

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

4645 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)(11,17)(12,23)(13,18)(14,22)(15,20)(16,21)(19,24);(2,6)(3,8)(9,11,16,15)(10,17,21,20)(12,19,13,22)(14,23,24,18)(25,26)(27,34)(28,36)(29,31)(30,33)(32,37)(35,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12,16,13)(10,18,21,23)(11,22,15,19)(14,17,24,20)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,39,36,40);(1,3)(2,4)(5,8)(6,7)(9,13,16,12)(10,18,21,23)(11,22,15,19)(14,20,24,17)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(1,4)(2,3)(5,7)(6,8)(9,14,16,24)(10,19,21,22)(11,23,15,18)(12,17,13,20)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,37)(38,40);(9,15,16,11)(10,20,21,17)(12,19,13,22)(14,23,24,18)(25,29,30,27)(26,31,33,34)(28,35,37,38)(32,40,36,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8607571);