#512304

order	512
nilpotency class	4
isoclinism	isoclinic class of #512072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,3)(5,6,9,13)(7,12,17,34)(8,21,22,56)(10,27,25,60)(11,31,26,16)(14,39,37,23)(15,41,38,30)(18,47,45,55)(19,50,28,61)(20,53,46,36)(24,43,32,57)(29,40,59,44)(33,58,64,48)(35,63,42,52)(49,66,54,68)(51,65,62,67);(1,2,4,3)(5,7)(6,12)(8,18)(9,17)(10,28)(11,20)(13,34)(14,33)(15,42)(16,36)(19,25)(21,55)(22,45)(23,58)(24,49)(26,46)(27,61)(29,62)(30,63)(31,53)(32,54)(35,38)(37,64)(39,48)(40,65)(41,52)(43,68)(44,67)(47,56)(50,60)(51,59)(57,66);(6,13)(7,17)(10,25)(14,37)(16,31)(18,45)(20,46)(23,39)(29,59)(30,41)(35,42)(40,44)(47,55)(49,54)(50,61)(66,68);(1,3,4,2)(5,8,11,32,9,22,26,24)(6,14,16,44,13,37,31,40)(7,18,20,54,17,45,46,49)(10,29,30,39,25,59,41,23)(12,33,36,67,34,64,53,65)(15,43,27,56,38,57,60,21)(19,51,52,58,28,62,63,48)(35,66,50,55,42,68,61,47)

Polycyclic presentation:

2^9|a^2=i,b^2=efi,d^2=eh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=ghi,h^d=hi

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, 419500);