#470938

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16718
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

686 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=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, 378134);