#6357304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357300
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

2436 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264500);