#483627

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×AGL(1,ℤ/8)	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dg,e^a=eg,c^b=ci,e^b=eg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	M₃₂×AGL(1,ℤ/8), #72342595, #72342596
maximal normal subgroup of	M₃₂×AGL(1,ℤ/8), #72342595, #72342596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390823);