#8914431

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	#50685	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4425 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,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,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, 8821627);