#93651

order	512
nilpotency class	4
isoclinism	isoclinic class of #93600
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1307
commutator	C₂²×C₈	C₄²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7049
prime radical	C₂³	#338
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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=fgh,e^2=h,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, 847);