#470934

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
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

696 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi,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, 378130);