#200409

order	512
nilpotency class	3
isoclinism	isoclinic class of #200406
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1107 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:

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

Polycyclic presentation:

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

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, 107605);