#484692

order	512
nilpotency class	4
isoclinism	isoclinic class of #484606
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#392	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

868 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391888);