#251673

order	512
nilpotency class	3
isoclinism	isoclinic class of #251671
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1067 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1067 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,e^c=eh,f^c=fi,g^c=gh,f^d=fh

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, 158869);