#149067

order	512
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56263);