#512029

order	512
nilpotency class	4
isoclinism	isoclinic class of #511802
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	C₂²×Q₈	C₂⁴
prime radical	C₂³	C₂×#120
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

862 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,c^b=ci,d^c=di,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271508
maximal normal subgroup of	#72271508
maximal quotient of	#72271531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419225);