#130673

order	512
nilpotency class	4
isoclinism	isoclinic class of #130510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2357 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=ei,f^b=fghi,d^c=di,e^c=ei,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37869);