#149171

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

464 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,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, 56367);