#506300

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

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

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, 413496);