#263404

order	512
nilpotency class	3
isoclinism	isoclinic class of #263403
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 128

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

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

Polycyclic presentation:

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

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, 170600);