#134397

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^c=egi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41593);