#317958

order	512
nilpotency class	3
isoclinism	isoclinic class of #317945
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	#555
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

489 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225154);