#6357113

order	512
nilpotency class	3
isoclinism	isoclinic class of #19621
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2608 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:

(2,5)(3,6)(7,8)(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);(1,2,6,8,4,7,3,5);(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,c^2=gi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₈×#575, #72286454, #72286509
maximal normal subgroup of	D₈×#575, #72286454, #72286509

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264309);