#6347344

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347305
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285515
maximal normal subgroup of	#72285515

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254540);