#93601

order	512
nilpotency class	4
isoclinism	isoclinic class of #93600
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1307
commutator	C₂²×C₈	C₄²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7040
prime radical	C₂³	H(3,ℤ/4)
LCS	G ▷ C₂²×C₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,10,7,12,27,40)(3,14,16,44)(4,19,22,38,23,9,26,35)(6,11,28,42,21,13,43,39)(8,29,30,52)(15,31,47,61)(17,34,50,59)(18,51,25,55)(20,33,49,37)(24,36,54,63,53,41,58,62)(32,57,60,48)(45,56,64,46);(1,3)(2,8,11,31,12,32,13,33)(4,20,23,15)(5,25,27,56)(6,18)(7,17)(9,34,36,51,19,14,41,45)(10,37,39,60,40,61,42,30)(16,28,50,43)(21,46)(22,29)(24,52,53,48)(26,57)(35,55,62,59,38,64,63,44)(47,54)(49,58)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 797);