#8429525

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
prime radical	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8336721);