#311336

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=ei,d^2=fhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 218532);