#419833

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1669 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,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, 327029);