#6353853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60646	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1897 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)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24);(9,10,12,14,13,15,19,20)(11,16,18,21,17,22,23,24);(1,2,4,6)(3,8,7,5)(10,15)(14,20)(16,22)(21,24);(9,11)(10,16)(12,18)(13,17)(14,21)(15,22)(19,23)(20,24);(11,17)(16,22)(18,23)(21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261049);