#381082

order	512
nilpotency class	3
isoclinism	isoclinic class of #380391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2331
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14286
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

639 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)(3,12,16,29)(4,17)(5,10)(6,22)(7,11)(8,14,24,15)(9,21)(13,25,31,28)(18,23,30,32)(19,27)(20,26);(1,3,6,15,7,16,5,14)(2,8,10,12,11,24,22,29)(4,18,19,31,21,30,20,13)(9,25,27,32,17,28,26,23);(1,4)(2,9,11,17)(3,13)(5,19)(6,20)(7,21)(8,23,24,32)(10,26,22,27)(12,28,29,25)(14,18)(15,30)(16,31);(4,19,21,20)(9,26,17,27)(13,18,31,30)(23,28,32,25)

Polycyclic presentation:

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

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, 288278);