#413125

order	512
nilpotency class	3
isoclinism	isoclinic class of #404632
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

986 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,c^2=e,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi

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, 320321);