#8810314

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	#46318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4278 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717510);