#330940

order	512
nilpotency class	3
isoclinism	isoclinic class of #330923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

601 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(9,10,16,21)(11,22,25,17)(12,28,33,47)(13,19,35,46)(14,36,15,20)(18,32,43,31)(23,51,53,41)(24,49,56,39)(26,40,27,50)(29,38,54,48)(30,59,62,69)(34,45,37,44)(42,63,70,61)(52,71,55,68)(57,66,58,67)(60,64,72,65);(1,2,4,7)(3,5,8,6)(9,11,14,26,16,25,15,27)(10,17,20,40,21,22,36,50)(12,29,31,23,33,54,32,53)(13,24,34,57,35,56,37,58)(18,41,28,48,43,51,47,38)(19,39,44,66,46,49,45,67)(30,60,61,52,62,72,63,55)(42,68,59,65,70,71,69,64);(2,5)(3,8)(6,7)(9,12,16,33)(10,18,21,43)(11,23,25,53)(13,30,35,62)(14,32,15,31)(17,38,22,48)(19,42,46,70)(20,47,36,28)(24,55,56,52)(26,29,27,54)(34,63,37,61)(39,65,49,64)(40,51,50,41)(44,69,45,59)(57,72,58,60)(66,68,67,71);(9,13)(10,19)(11,24)(12,30)(14,34)(15,37)(16,35)(17,39)(18,42)(20,44)(21,46)(22,49)(23,52)(25,56)(26,57)(27,58)(28,59)(29,60)(31,61)(32,63)(33,62)(36,45)(38,64)(40,66)(41,68)(43,70)(47,69)(48,65)(50,67)(51,71)(53,55)(54,72)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238136);