#200904

order	512
nilpotency class	3
isoclinism	isoclinic class of #200746
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60803	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1215 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338226, #72338248
maximal normal subgroup of	#72338226, #72338248

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108100);