#470888

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16708
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

708 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=fh,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 378084);