#94786

order	512
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1404
commutator	#125	C₄²
Frattini, cosocle	#3135	C₂²
socle	C₂	#7489
prime radical	#532	D₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3135 ◁ G
LECS	G ▷ #3135 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

166 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=df,f^b=fhi,g^b=gh,d^c=dgh,f^c=fh,g^c=gi,f^d=fhi,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1982);