#6357897

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357891
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2225 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)(3,7)(6,8)(9,10)(11,13)(12,14)(15,18)(16,22)(17,19)(20,21)(23,24);(9,11,19,23,15,21,14,22)(10,16,12,20,18,24,17,13);(1,2,3,5,4,6,7,8)(9,12,19,18,15,17,14,10)(11,20,23,24,21,13,22,16);(2,6)(5,8)(9,13)(10,11)(12,23)(14,24)(15,20)(16,19)(17,22)(18,21);(11,21)(13,20)(16,24)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265093);