#484179

order	512
nilpotency class	4
isoclinism	isoclinic class of #484176
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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	#23406	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1126 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1126 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,17)(13,16)(14,19)(18,27)(20,29)(21,28)(22,31)(23,33)(24,32)(25,35)(26,30)(34,38)(36,39)(37,40);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,19,31)(12,23,25,20)(13,26,27,21)(16,28,18,30)(17,33,35,29)(24,36,38,37)(32,40,34,39);(10,16)(13,27)(15,28)(17,32)(18,19)(21,26)(24,38)(29,39)(30,31)(33,40)(34,35)(36,37);(9,12,14,25)(10,17,19,35)(11,20,22,23)(13,24,27,38)(15,29,31,33)(16,32,18,34)(21,36,26,37)(28,39,30,40)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,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, 391375);