#455170

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6)(7,8)(10,11)(13,14,17,21,16,20,25,27)(15,22,24,28,23,18,26,19);(1,2)(3,4)(9,10,12,11)(13,15,16,23)(14,18,20,22)(17,24,25,26)(19,27,28,21);(2,4)(9,10)(11,12)(13,15)(14,19)(16,23)(17,26)(18,21)(20,28)(22,27)(24,25);(6,8)(9,11,12,10)(15,23)(18,22)(19,28)(24,26)

Polycyclic presentation:

2^9|a^2=efi,b^2=e,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362366);