#116315

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1584 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)(13,15)(17,18)(19,21)(20,23)(22,24);(9,10)(11,15,16,13)(12,14)(17,18)(19,21)(20,23)(22,24);(2,4)(9,11)(10,13)(12,16)(14,15)(18,21)(19,24)(22,23)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ehi,f^b=fhi,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215419, #72217661, #72267816
maximal normal subgroup of	#72217661, #72267816
maximal quotient of	#72215344, #72215369, #72215373

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23511);