#205867

order	512
nilpotency class	3
isoclinism	isoclinic class of #9781
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

965 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)(3,13)(4,16)(5,10)(6,11)(7,12)(8,14)(9,15)(17,18,22,26,21,24,20,25)(19,23,27,31,28,30,29,32);(1,3,5,14,7,15,6,4)(2,8,11,13,12,16,10,9)(17,19,22,29,21,28,20,27)(18,23,25,31,24,30,26,32);(1,4,6,15,7,14,5,3)(2,9,10,16,12,13,11,8)(18,24)(23,30)(25,26)(31,32);(1,5,7,6)(2,10,12,11)(3,14,15,4)(8,9,16,13)(17,20,21,22)(18,25,24,26)(19,27,28,29)(23,31,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113063);