#470939

order	512
nilpotency class	3
isoclinism	isoclinic class of #457608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2735
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

673 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 378135);