#8614120

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	#43015	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) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=gi,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, 8521316);