#10195937

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
prime radical	#44513	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4218 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=ghi,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, 10103133);