#479755

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 749 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eh,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386951);