#135948

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8851
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

297 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,g^a=gi,d^b=di,e^b=egi,d^c=dg,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287454, #72343179, #72343187
maximal normal subgroup of	#72287454, #72343179, #72343187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43144);