#135949

order	512
nilpotency class	4
isoclinism	isoclinic class of #2204
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

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:

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

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,c^b=ci,d^b=di,e^b=egi,d^c=dg,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287453, #72343179, #72343182
maximal normal subgroup of	#72287453, #72343179, #72343182

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43145);