#474246

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16926
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

619 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:

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

Polycyclic presentation:

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

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, 381442);