#6354300

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354299
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|b^2=fh,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,g^c=gi,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, 6261496);