#500114

order	512
nilpotency class	4
isoclinism	isoclinic class of #500112
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

966 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fhi,h^c=hi,f^d=fi

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, 407310);