#129834

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1782
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

613 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 613 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fi,f^d=fi

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, 37030);