#471229

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16780
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

631 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)(4,16,20,32)(5,10)(6,24,25,12)(9,26,29,19)(13,18)(14,28)(15,27)(17,21)(22,31)(23,30);(1,3,7,13)(2,8)(4,17)(5,9,23,29)(6,15,25,14)(10,26)(11,18)(12,28)(16,21,32,22)(19,30)(20,31)(24,27);(1,4,6,19,7,20,25,26)(2,9,12,16,11,29,24,32)(3,10,14,31,13,30,15,17)(5,18,21,28,23,8,22,27);(1,5,6,21,7,23,25,22)(2,10,24,17,11,30,12,31)(3,9,15,32,13,29,14,16)(4,18,19,28,20,8,26,27)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=f,d^2=ef,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, 378425);