#8703159

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8610355);