#512156

order	512
nilpotency class	4
isoclinism	isoclinic class of #511785
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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	#23904	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1191 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1191 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24);(2,5)(6,8)(9,11,13,17)(10,14,16,21)(12,18,19,24)(15,22,23,20);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(14,20)(16,23)(18,24)(21,22);(11,17)(12,19)(14,21)(20,22)

Polycyclic presentation:

2^9|b^2=i,c^2=e,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, 419352);