#135674

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#107
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26)(27,39)(28,33)(29,47)(30,50)(31,51)(32,38)(34,43)(35,40)(36,49)(37,48)(41,53)(42,52)(44,56)(45,55)(46,54);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,40,45,31,42,49,46,32,43,30,41,48,44,29,28)(26,33,47,56,37,53,50,34,38,54,36,52,51,55,35,39);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28,29,44,48,41,30,43,32,46,49,42,31,45,40,27)(26,34,35,53,51,56,36,33,38,39,50,55,37,52,47,54)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=d,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42870);