#8657131

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	#56346	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8564327);