#199897

order	512
nilpotency class	3
isoclinism	isoclinic class of #199824
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1097 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107093);