#199114

order	512
nilpotency class	3
isoclinism	isoclinic class of #198816
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	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

1267 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257626, #72257630, #72257636, …
maximal normal subgroup of	#72257626, #72257630, #72257636, …
maximal quotient of	#72257950, #72264135

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106310);