#499363

order	512
nilpotency class	4
isoclinism	isoclinic class of #499054
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

718 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 218 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406559);