#317929

order	512
nilpotency class	3
isoclinism	isoclinic class of #317914
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#537
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

495 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=i,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225125);