#484252

order	512
nilpotency class	4
isoclinism	isoclinic class of #484162
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×C₂²⋊C₄	C₂⁴
prime radical	#392	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 945 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)(11,16)(12,17)(13,15)(14,18)(19,30)(20,27)(21,31)(22,32)(23,28)(24,33)(25,37)(26,29)(34,38)(35,39)(36,40);(2,5)(3,8)(6,7)(10,15)(16,27)(17,28)(18,29)(30,38)(31,37)(32,39)(33,40);(9,11,14,21)(10,16,18,31)(12,22,24,19)(13,25,26,20)(15,27,29,37)(17,32,33,30)(23,34,36,35)(28,39,40,38);(9,12,14,24)(10,17,18,33)(11,19,21,22)(13,23,26,36)(15,28,29,40)(16,30,31,32)(20,34,25,35)(27,38,37,39)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,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, 391448);