#8841347

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	#36263	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8748543);