#135957

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
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×D₄	C₂²×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8851
prime radical	#525	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 301 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287453, #72343185, #72343187
maximal normal subgroup of	#72287453, #72343185, #72343187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43153);