#9017366

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	#51581	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4353 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8924562);