#8110564

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	#39818	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017760);