#149355

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#481
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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)(9,10,14,19,24,22,12,17,13,18,23,21,15,11,20,16);(2,4)(5,6)(7,8)(9,11,15,10,13,22,24,18)(12,17,14,21,20,16,23,19);(2,4)(6,8)(9,10,15,22,13,18,24,11)(12,19,14,17,20,21,23,16)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=ehi,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,g^c=ghi,h^c=hi

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, 56551);