#152195

order	512
nilpotency class	5
isoclinism	isoclinic class of #152194
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12759
commutator	C₄⋊Q₈	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#61111	C₂
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

801 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,14,17,8)(4,19,22,39)(5,11)(6,12,30,10)(7,13)(9,21,35,24)(15,45,46,64)(16,44,50,32)(18,33,27,34)(20,55)(23,59,51,38)(25,37,29,40)(26,41)(28,54)(31,47,63,57)(36,58)(42,56)(43,60,61,49)(48,53,52,62);(1,3,7,18)(2,8,13,34)(4,15,22,48)(5,23,26,50)(6,27,30,17)(9,31,35,61)(10,33,12,14)(11,38,41,44)(16,29,51,25)(19,53,39,45)(20,52,28,46)(21,57,24,60)(32,37,59,40)(36,63,42,43)(47,58,49,56)(54,62,55,64);(1,4)(2,9)(3,15)(5,24)(6,28)(7,22)(8,31)(10,36)(11,39)(12,42)(13,35)(14,43)(16,49)(17,52)(18,48)(19,41)(20,30)(21,26)(23,60)(25,58)(27,46)(29,56)(32,64)(33,63)(34,61)(37,54)(38,45)(40,55)(44,53)(47,51)(50,57)(59,62)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59391);