#93960

order	512
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1343
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	#1729	C₂²
socle	C₂	#7123
prime radical	C₂⁴	#103
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

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

normal subgroups only

This group has 224 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)(3,15,17,45,19,46,26,57)(4,20,22,51)(6,12,29,43)(7,13,27,41)(8,14,28,42)(9,31,32,16,34,47,61,48)(10,35,37,24)(18,36,50,62,30,58,49,64)(21,44,52,59,53,33,54,60)(23,39,56,40)(25,38,55,63);(1,3,7,30,8,19,6,18)(2,9,12,33,14,34,13,44)(4,21,24,16,25,53,23,48)(5,26,29,49,28,17,27,50)(10,36,20,46,39,58,38,15)(11,32,41,59,42,61,43,60)(22,54,56,47,55,52,35,31)(37,62,63,45,40,64,51,57)

Polycyclic presentation:

2^9|a^2=d,b^2=eh,c^2=fh,d^2=g,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defh,f^b=fi,g^b=gi,d^c=dhi,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, 1156);