#412097

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319293);