#200225

order	512
nilpotency class	3
isoclinism	isoclinic class of #200152
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	#60790	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1103 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257478
maximal normal subgroup of	#72257478
maximal quotient of	#72257678

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107421);