#8701272

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

4405 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=ghi,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, 8608468);