#136628

order	512
nilpotency class	4
isoclinism	isoclinic class of #136628
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8943
prime radical	#3050	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

371 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,13,17,41,19,40,18,36)(4,20,23,9)(5,10)(6,12)(7,11)(8,16,31,47,30,48,32,25)(14,28)(15,44,46,63,21,39,49,64)(22,50,56,34)(24,51,55,35)(26,37)(27,38)(29,45,60,57,33,53,61,54)(42,62,52,58)(43,59);(1,3,14,16,6,17,27,47,7,19,43,48,26,18,5,25)(2,8,28,13,12,31,38,41,11,30,59,40,37,32,10,36)(4,21,52,45,24,46,22,54,23,15,42,53,55,49,56,57)(9,33,62,39,35,60,34,63,20,29,58,44,51,61,50,64);(1,4)(2,9)(3,15)(5,22)(6,24)(7,23)(8,29)(10,34)(11,20)(12,35)(13,39)(14,42)(16,45)(17,49)(18,46)(19,21)(25,57)(26,55)(27,56)(28,58)(30,33)(31,61)(32,60)(36,64)(37,51)(38,50)(40,44)(41,63)(43,52)(47,54)(48,53)(59,62)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,d^b=dg,f^b=fg,g^b=gi,h^b=hi,f^c=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, 43824);