#512895

order	512
nilpotency class	5
isoclinism	isoclinic class of #512894
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂⁴
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1009 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1009 conjugacy classes of subgroups — too many to draw. Only its 112 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gh,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,f^c=fi,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420091);