#512016

order	512
nilpotency class	4
isoclinism	isoclinic class of #511785
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×D₄	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 994 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,23)(22,24);(1,3,4,8)(2,5,6,7)(11,12)(15,23)(16,19)(20,21);(9,11,14,12)(10,15,18,23)(13,20,22,21)(16,17,19,24);(9,12,14,11)(10,16,18,19)(13,20,22,21)(15,17,23,24)

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,c^b=ci,d^b=di,g^c=ghi,f^d=fh,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419212);