#200729

order	512
nilpotency class	3
isoclinism	isoclinic class of #200728
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	#60968	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1143 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=gi,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	#72316632, #72316635
maximal normal subgroup of	#72316632, #72316635

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107925);