#512013

order	512
nilpotency class	4
isoclinism	isoclinic class of #511775
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, prime radical, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

923 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 923 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282351, #72282516, #72282650
maximal normal subgroup of	#72282351, #72282516, #72282650

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419209);