#10168383

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

4464 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:

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

Polycyclic presentation:

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

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, 10075579);