#8697514

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

4392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=ghi,f^2=g,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, 8604710);