#471170

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16772
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

630 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)(3,8,14,22)(4,15,17,26)(5,9,19,24)(6,11)(7,10)(12,20)(13,21)(16,29)(18,25,31,27)(23,32)(28,30);(1,3,7,14)(2,8)(4,12)(5,13,19,23)(6,15,20,26)(9,21)(10,22)(11,17)(16,27)(18,28,31,29)(24,32)(25,30);(1,4,6,8,7,17,20,22)(2,3,12,26,10,14,11,15)(5,16,18,21,19,30,31,32)(9,23,27,28,24,13,25,29);(1,5,6,18,7,19,20,31)(2,9,11,25,10,24,12,27)(3,13,15,28,14,23,26,29)(4,16,8,21,17,30,22,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,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, 378366);