#8761748

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	#56339	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4566 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

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, 8668944);