#153204

order	512
nilpotency class	5
isoclinism	isoclinic class of #153203
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13348
commutator, Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂×#454	C₂²
LCS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279336, #72279869, #72279992, …
maximal normal subgroup of	#72279336, #72279869, #72279992, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60400);