#6629393

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628892
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

4325 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252419, #72252427, #72252430, …
maximal normal subgroup of	#72252419, #72252427, #72252430, …
maximal quotient of	#72267135

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536589);