#418828

order	512
nilpotency class	3
isoclinism	isoclinic class of #417696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1906 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326024);