#10166172

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	#49584	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=h,e^2=ghi,f^2=gh,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, 10073368);