#289407

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#540
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

511 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,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,c^b=cg,d^b=dh,g^b=ghi,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196603);