#419902

order	512
nilpotency class	3
isoclinism	isoclinic class of #418504
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	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1580 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=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, 327098);