#139962

order	512
nilpotency class	4
isoclinism	isoclinic class of #139962
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#2397	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

551 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=g,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,e^b=ehi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,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, 47158);