#201623

order	512
nilpotency class	3
isoclinism	isoclinic class of #201394
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	C₂×C₄²	C₂⁴
prime radical	#2433	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1289 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,f^b=fi,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, 108819);