#199864

order	512
nilpotency class	3
isoclinism	isoclinic class of #9345
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3006	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107060);