#199141

order	512
nilpotency class	3
isoclinism	isoclinic class of #198834
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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1199 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,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	#72257616, #72257646, #72257675
maximal normal subgroup of	#72257616, #72257646, #72257675
maximal quotient of	#72240832, #72264279

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106337);