#411621

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1796 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)(7,19)(8,24)(9,29)(10,17)(11,16)(12,18)(13,21)(14,27)(15,30)(20,36)(22,32)(23,31)(25,35)(26,34)(28,33);(2,4)(5,7)(6,13)(8,25)(9,28)(10,22)(11,21)(12,23)(14,29)(15,33)(16,31)(17,19)(18,32)(20,27)(24,36)(26,30)(34,35);(5,8)(6,14)(7,20)(9,22)(10,27)(11,26)(12,28)(13,15)(16,24)(17,33)(18,34)(19,35)(21,25)(23,30)(29,31)(32,36);(5,9,10,30,12,25,11,20)(6,15,16,29,18,36,17,35)(7,8,22,27,23,28,21,26)(13,24,31,34,32,33,19,14)

Polycyclic presentation:

2^9|d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 318817);