#500209

order	512
nilpotency class	4
isoclinism	isoclinic class of #499873
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	#59056	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1170 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 134 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)(10,15)(11,12)(13,21)(16,23)(17,19)(22,24);(2,5)(6,8)(10,16)(13,21)(15,23)(17,22)(18,20)(19,24);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,19)(13,22)(14,17)(16,21)(18,24)(20,23);(2,5)(6,8)(9,11,14,12)(10,15,17,19)(13,18,21,20)(16,23,22,24)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407405);