#153132

order	512
nilpotency class	5
isoclinism	isoclinic class of #153132
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13343
commutator, Frattini, cosocle	C₂×#124	C₂³
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

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

normal subgroups only

This group has 590 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)(3,13)(4,9)(5,21)(6,11)(7,12)(8,15)(10,24)(14,45)(16,38)(17,44)(18,54)(19,36)(20,37)(22,30)(23,58)(25,40)(26,35)(27,41)(28,42)(29,59)(31,53)(32,50)(33,46)(34,56)(39,57)(43,47)(48,60)(49,61)(51,63)(52,64)(55,62);(1,3)(2,8)(4,14)(5,22)(6,17)(7,16)(9,33)(10,38)(11,31)(12,30)(13,21)(15,24)(18,55)(19,52)(20,47)(23,49)(25,53)(26,48)(27,50)(28,51)(29,35)(32,42)(34,43)(36,61)(37,62)(39,64)(40,63)(41,44)(45,54)(46,56)(57,59)(58,60);(1,4)(2,9)(3,14)(5,23)(6,26)(7,20)(8,29)(10,39)(11,35)(12,37)(13,43)(15,49)(16,47)(17,52)(18,25)(19,28)(21,58)(22,59)(24,57)(27,56)(30,61)(31,33)(32,62)(34,41)(36,42)(38,45)(40,54)(44,60)(46,50)(48,51)(53,55)(63,64)

Polycyclic presentation:

2^9|d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fgh,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=gh,d^c=dg,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	#72286041, #72286052, #72286062
maximal normal subgroup of	#72286041, #72286052, #72286062

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60328);