#349946

order	512
nilpotency class	3
isoclinism	isoclinic class of #346582
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,f^c=fh,g^c=gh,f^d=fh,g^d=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257142);