#8970271

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	#59176	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8877467);