#384894

order	512
nilpotency class	3
isoclinism	isoclinic class of #31414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14392
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

822 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=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, 292090);