#93640

order	512
nilpotency class	4
isoclinism	isoclinic class of #93511
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1297
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×#405	C₂²
socle	C₂	#7046
prime radical	C₂⁴	#108
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

197 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=fh,e^2=h,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efghi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,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, 836);