#384926

order	512
nilpotency class	3
isoclinism	isoclinic class of #31282
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14393
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,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, 292122);