#451497

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(5,6,12,16)(7,11,15,10)(8,19)(9,18,26,29)(13,21)(14,24,30,25)(17,23)(20,27)(22,28)(31,33)(32,34)(35,36);(5,7,11,6,12,15,10,16)(8,17,22,13,23,19,21,28)(9,14,25,29,26,30,24,18)(20,27,34,32,35,36,33,31);(5,8)(6,13,16,28)(7,17,15,19)(9,20)(10,21)(11,22)(12,23)(14,27,30,36)(18,31,29,32)(24,33)(25,34)(26,35);(2,4)(5,9,10,24,12,26,11,25)(6,14,15,29,16,30,7,18)(8,20,21,33,23,35,22,34)(13,27,19,32,28,36,17,31)

Polycyclic presentation:

2^9|a^2=egh,b^2=f,c^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358693);