#381417

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14298
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

675 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:

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

Polycyclic presentation:

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

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, 288613);