#6454874

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62781	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=gi,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310926
maximal normal subgroup of	#72310926
maximal quotient of	#72313473

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362070);