#149437

order	512
nilpotency class	4
isoclinism	isoclinic class of #129430
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1782
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

897 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 897 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,g^b=ghi,d^c=di,g^c=gi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56633);