#152563

order	512
nilpotency class	5
isoclinism	isoclinic class of #152563
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13041
commutator	C₄⋊Q₈	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#3018	C₂²
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

511 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 60 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)(4,21,24,40)(5,27)(6,11)(7,13,32,44)(8,14)(9,17)(10,23,37,22)(12,31,41,28)(16,47,49,61)(18,35,53,63)(19,42,55,46)(20,34)(25,58,26,43)(29,36)(30,38,57,39)(33,50,60,51)(45,48,56,52)(54,64,59,62);(1,3,5,17,8,20,6,29)(2,9,11,15,14,36,27,34)(4,16,23,52,26,51,30,59)(7,19,28,53,32,55,31,18)(10,33,40,45,39,47,58,62)(12,42,13,35,41,46,44,63)(21,56,38,61,43,64,37,60)(22,48,25,50,57,54,24,49);(1,4)(2,10)(3,16)(5,23)(6,30)(7,24)(8,26)(9,33)(11,40)(12,43)(13,37)(14,39)(15,45)(17,52)(18,54)(19,49)(20,51)(21,41)(22,28)(25,32)(27,58)(29,59)(31,57)(34,62)(35,60)(36,47)(38,44)(42,64)(46,56)(48,53)(50,55)(61,63)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=eghi,h^b=hi,d^c=dg,e^c=ei,f^c=fg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209895
maximal normal subgroup of	#72209895
maximal quotient of	#72227839, #72228011, #72228034

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59759);