#10089583

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	#42119	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4213 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,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, 9996779);