#8873735

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

4539 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8780931);