#375668

order	512
nilpotency class	3
isoclinism	isoclinic class of #213728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1295 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1295 conjugacy classes of subgroups — too many to draw. Only its 250 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)(3,7)(4,5)(6,8)(9,10,13,17,14,11,15,19)(12,18,21,24,20,16,22,23);(2,5)(6,7)(10,11)(13,15)(16,18)(21,22);(9,11,13,19,14,10,15,17)(12,18,22,23,20,16,21,24);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,18)(13,22)(14,20)(15,21)(17,24)(19,23)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282864);