#119388

order	512
nilpotency class	3
isoclinism	isoclinic class of #119386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

632 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,e^c=ei,g^c=ghi

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, 26584);