#151945

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂×C₄²
Frattini, prime radical, cosocle	#532	C₂³
socle	C₂	#12627
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ch,d^a=dh,f^a=fgh,g^a=gi,d^b=dg,g^b=gh,h^b=hi,d^c=di,f^d=fhi,g^f=gi

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, 59141);