#470933

order	512
nilpotency class	3
isoclinism	isoclinic class of #462392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2517
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

695 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,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, 378129);