#387199

order	512
nilpotency class	3
isoclinism	isoclinic class of #232138
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

698 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 226 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)(3,7)(9,10,15,19,14,20,13,23)(11,17,12,16,21,18,22,24);(1,2,4,5)(3,8,7,6)(9,11,14,21)(10,16,20,24)(12,13,22,15)(17,19,18,23);(9,11,15,22,14,21,13,12)(10,17,19,24,20,18,23,16);(1,3,4,7)(2,6,5,8)(9,12,13,21,14,22,15,11)(10,18,19,16,20,17,23,24)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=g,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294395);