#387378

order	512
nilpotency class	3
isoclinism	isoclinic class of #384777
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2337
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14469
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=efg,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294574);