#201441

order	512
nilpotency class	3
isoclinism	isoclinic class of #201434
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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

1136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108637);