#6353633

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2273 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)(3,5)(4,7)(6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(9,11,13,21,14,19,12,20)(10,15,16,18,17,23,22,24);(2,5)(3,8)(6,7);(11,19)(15,23)(18,24)(20,21);(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285399, D₈×#575, #72286469
maximal normal subgroup of	#72285399, D₈×#575, #72286469
maximal quotient of	#72285618, #72319626

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260829);