#147200

order	512
nilpotency class	4
isoclinism	isoclinic class of #12087
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319737, #72319745, #72319763, …
maximal normal subgroup of	#72319737, #72319745, #72319763, …
maximal quotient of	#72322531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54396);