#106424

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1009 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,21,24,22,25,20,26)(19,23,28,30,27,31,29,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23,25,31)(20,28)(21,29)(22,27)(24,32,26,30);(3,9)(6,12)(10,15)(13,16)(18,24)(19,27)(20,21)(23,30)(25,26)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ci,d^b=dh,f^b=fh,e^c=eh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13620);