#10089567

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	#50024	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4328 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)(11,19,20,16)(12,17,22,21)(13,23,24,18)(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)(12,22)(16,19)(18,23)(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)(11,20)(17,21)(18,23)(26,33)(27,36)(28,38)(35,40);(1,2)(3,5)(4,7)(6,8)(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,28)(26,34)(27,37)(29,36)(30,33)(31,38)(32,40)(35,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,16,24,19)(25,29,31,37)(26,35,33,40)(27,38,36,28)(30,39,34,32);(1,2)(3,5)(4,7)(6,8)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(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=h,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, 9996763);