#10349008

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

4917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 10256204);