#201200

order	512
nilpotency class	3
isoclinism	isoclinic class of #201085
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108396);