#8617753

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

4389 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=gh,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, 8524949);