#146499

order	512
nilpotency class	4
isoclinism	isoclinic class of #12725
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂⁴	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#10726
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

801 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,f^b=fh,e^c=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53695);