#8110554

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4278 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)(10,15)(12,20)(13,16)(19,24)(25,26,32,36)(27,30,34,29)(28,35,31,33)(37,39,38,40);(2,5)(6,8)(10,15)(11,18)(12,20)(21,23)(25,27,32,34)(26,29,36,30)(28,37,31,38)(33,39,35,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,16,18,13)(12,17,20,22)(19,23,24,21)(25,28,32,31)(26,33,36,35)(27,37,34,38)(29,39,30,40);(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,29,32,30)(26,34,36,27)(28,39,31,40)(33,38,35,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,30,32,29)(26,27,36,34)(28,40,31,39)(33,37,35,38);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,11,15,18)(12,21,20,23)(17,19,22,24)(25,31,32,28)(26,35,36,33)(27,38,34,37)(29,40,30,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=g,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, 8017750);