#427427

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1203 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^d=gi,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, 334623);