#128216

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

274 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)(27,39,42,51)(28,34)(29,35)(30,47)(31,37)(32,38)(33,41,49,44)(36,45)(40,50,55,56)(43,48,54,53)(46,52);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,43,31,44,46,55,32,42,45,54,29,41,28,40)(26,33,36,50,37,51,34,48,38,49,47,56,35,39,52,53);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28,29,45,32,46,31,30)(26,34,35,36,38,52,37,47)(27,40,41,54,42,55,44,43)(33,48,39,50,49,53,51,56)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,f^a=fi,c^b=cg,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35412);