#279269

order	512
nilpotency class	3
isoclinism	isoclinic class of #267051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

683 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=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, 186465);