#10328250

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

4770 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=h,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, 10235446);