#427305

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1432 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:

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

Polycyclic presentation:

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

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, 334501);