#153095

order	512
nilpotency class	5
isoclinism	isoclinic class of #153086
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33607	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

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

normal subgroups only

This group has 363 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,g^b=gh,h^b=hi,d^c=di,e^c=ei

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, 60291);