#391985

order	512
nilpotency class	3
isoclinism	isoclinic class of #391902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14668
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

623 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)(3,11)(4,15,18,25)(5,9)(6,10)(7,14)(8,17,24,19)(12,28,16,32)(13,27)(20,26)(21,30,23,29)(22,31);(1,3)(2,7,10,11)(4,16)(5,13)(6,14)(8,23,24,21)(9,22,26,27)(12,18)(15,28,25,32)(17,30)(19,29)(20,31);(1,4)(2,8)(3,12,14,16)(5,19)(6,18)(7,21,11,23)(9,25)(10,24)(13,30,31,29)(15,26)(17,20)(22,28,27,32);(3,13,14,31)(7,22,11,27)(12,29,16,30)(21,32,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299181);