#199158

order	512
nilpotency class	3
isoclinism	isoclinic class of #198843
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

1167 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=hi,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=dhi,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	#72257890, #72257895, #72257912
maximal normal subgroup of	#72257890, #72257895, #72257912
maximal quotient of	#72240832, #72264279

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106354);