#6371979

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279175);