#153313

order	512
nilpotency class	5
isoclinism	isoclinic class of #153081
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2218
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13448
prime radical	#33981	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

257 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,4)(5,24)(6,37)(7,41)(8,42)(9,18)(10,17)(11,19)(12,26)(13,34)(14,53)(15,32)(16,33)(20,59)(21,46)(22,54)(23,30)(25,31)(27,38)(28,43)(29,44)(35,57)(36,56)(39,58)(40,52)(45,49)(47,50)(48,51)(55,64)(60,63)(61,62);(1,3)(2,11)(4,24)(5,31)(6,38)(7,12)(8,43)(9,44)(10,23)(13,28)(14,54)(15,25)(16,26)(17,48)(18,27)(19,34)(20,35)(21,32)(22,33)(29,57)(30,56)(36,63)(37,46)(39,62)(40,51)(41,45)(42,47)(49,64)(50,53)(52,61)(55,60)(58,59);(1,4,5,25,32,46,6,27,9,29,35,59,39,61,40,48,10,30,36,60,64,45,7,26,33,54,53,47,8,28,34,11)(2,12,13,49,42,63,14,23,16,51,41,62,55,20,56,44,17,38,52,21,58,31,57,3,18,19,37,43,15,50,24,22)

Polycyclic presentation:

2^9|c^2=d,d^2=gh,e^2=gi,f^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fgh,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fghi,g^b=gh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320849, #72320861, #72320895
maximal normal subgroup of	#72320849, #72320861, #72320895

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60509);