#6357530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2484 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214347, #72283862, #72283889
maximal normal subgroup of	#72214347, #72283862, #72283889
maximal quotient of	#72224783

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264726);