#6454815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3095 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)(3,6)(4,5)(7,8)(9,10,16,19)(11,14,18,15)(12,21,13,17)(20,23,24,22);(9,11,15,10,16,18,14,19)(12,17,23,24,13,21,22,20);(2,5)(6,8)(9,12)(10,17)(11,20)(13,16)(14,22)(15,23)(18,24)(19,21);(9,13)(10,17)(11,20)(12,16)(14,23)(15,22)(18,24)(19,21);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,18,19,11)(12,22,13,23)(17,24,21,20)

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi

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, 6362011);