#153131

order	512
nilpotency class	5
isoclinism	isoclinic class of #153131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2199
commutator, Frattini, cosocle	#532	C₂³
socle	C₂	#13343
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 588 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)(3,15)(4,20)(5,12)(6,11)(7,13)(8,14)(9,17)(10,22)(16,32)(18,46)(19,40)(21,37)(23,56)(24,42)(25,34)(26,43)(27,39)(28,41)(29,59)(30,36)(31,44)(33,52)(35,53)(38,58)(45,48)(47,55)(49,62)(50,63)(51,61)(54,64)(57,60);(1,3)(2,9)(4,16)(5,25)(6,17)(7,29)(8,19)(10,35)(11,40)(12,15)(13,33)(14,34)(18,31)(20,55)(21,57)(22,48)(23,49)(24,50)(26,52)(27,51)(28,54)(30,53)(32,37)(36,61)(38,63)(39,62)(41,59)(42,60)(43,46)(44,64)(45,56)(47,58);(1,4)(2,10)(3,16)(5,22)(6,27)(7,30)(8,24)(9,32)(11,20)(12,42)(13,37)(14,39)(15,45)(17,51)(18,53)(19,50)(21,26)(23,31)(25,48)(28,58)(29,49)(33,35)(34,63)(36,41)(38,44)(40,61)(43,56)(46,60)(47,52)(54,57)(55,59)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209895, #72209901, #72209927
maximal normal subgroup of	#72209895
maximal quotient of	#72227847, #72228001, #72228035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60327);