#10102230

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

4408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 10009426);