#93914

order	512
nilpotency class	5
isoclinism	isoclinic class of #7410
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1344
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	#1729	C₂²
socle	C₂	#7104
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ #1729 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

210 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=f,d^2=g,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defi,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1110);