#116374

order	512
nilpotency class	3
isoclinism	isoclinic class of #116371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

731 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,e^c=ei,f^c=fh,g^c=gi

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, 23570);