#153084

order	512
nilpotency class	5
isoclinism	isoclinic class of #153083
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2221	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

355 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,12,19)(7,20,25,41)(8,13,30,38)(9,17,10,16)(11,32,33,47)(14,21,43,23)(15,44,46,26)(18,31,48,27)(22,54,58,39)(24,42,60,66)(28,36,29,35)(34,52,49,56)(37,62,45,61)(40,55,65,68)(50,67,64,57)(51,59,63,53);(6,13)(7,21)(8,26)(9,10)(11,27)(14,39)(15,18)(16,36)(17,35)(19,38)(20,49)(22,55)(23,25)(24,60)(28,62)(29,61)(30,44)(31,33)(32,37)(34,41)(40,63)(42,64)(43,54)(45,47)(46,48)(50,66)(51,65)(52,57)(56,67)(58,68);(2,4)(5,7)(6,14)(8,22)(9,21)(10,23)(11,24)(12,25)(13,34)(15,40)(16,41)(17,20)(18,42)(19,43)(26,57)(27,53)(28,52)(29,56)(30,58)(31,59)(32,63)(33,60)(35,39)(36,54)(37,50)(38,49)(44,67)(45,64)(46,65)(47,51)(48,66)(55,61)(62,68)

Polycyclic presentation:

2^9|a^2=i,d^2=gh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,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, 60280);