#347828

order	512
nilpotency class	3
isoclinism	isoclinic class of #347818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

906 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ch,d^b=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255024);