#108046

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

975 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=di,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279552, #72280613, #72280629
maximal normal subgroup of	#72279552, #72280613, #72280629

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15242);