#8751464

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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)(3,7)(4,8)(5,6)(9,10)(11,19)(12,13)(14,16)(15,18)(17,23)(20,21)(22,24)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(2,6)(3,8)(9,11,15,14)(10,16,18,19)(12,22,17,20)(13,21,23,24)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3)(2,4)(5,8)(6,7)(9,11,15,14)(10,16,18,19)(12,22,17,20)(13,21,23,24)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,17)(10,13,18,23)(11,20,14,22)(16,24,19,21)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,21)(12,18)(14,24)(15,23)(16,20)(19,22)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40);(9,14,15,11)(10,16,18,19)(12,20,17,22)(13,21,23,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658660);