#470737

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16697
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

763 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)(3,8,9,25)(4,16,18,15)(5,19)(7,13,22,12)(10,20)(14,29)(17,30)(21,27)(23,28)(24,26)(31,32);(1,3)(2,8,11,25)(4,13,18,12)(5,14)(6,9)(7,16)(10,24,19,29)(15,22)(17,27,32,28)(20,26)(21,31)(23,30);(1,4,7,8,6,18,22,25)(2,9,12,16,11,3,13,15)(5,17,21,29,20,32,23,24)(10,26,27,31,19,14,28,30);(1,5)(2,10)(3,14)(4,17)(6,20)(7,23)(8,24)(9,26)(11,19)(12,28)(13,27)(15,31)(16,30)(18,32)(21,22)(25,29)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 377933);