#10408073

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

4725 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:

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

Polycyclic presentation:

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

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, 10315269);