#428603

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15386
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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)(3,14,18,30)(4,9)(5,21)(6,11)(7,12)(8,17,27,19)(10,24)(13,25)(15,31,20,16)(22,28,26,32)(23,29);(1,3)(2,8)(4,20)(5,22)(6,19)(7,18)(9,26)(10,28)(11,30)(12,27)(13,14)(15,21)(16,23)(17,25)(24,31)(29,32);(1,4,6,10,7,21,25,29)(2,9,11,24,12,5,13,23)(3,15,17,28,18,20,19,32)(8,26,14,16,27,22,30,31);(1,5)(2,10)(3,16,18,31)(4,13)(6,23)(7,9)(8,15,27,20)(11,21)(12,29)(14,28,30,32)(17,22,19,26)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335799);