#9851158

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	#39942	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4194 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9758354);