#8334307

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8241503);