#6446737

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4096 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:

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

Polycyclic presentation:

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

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, 6353933);