#330935

order	512
nilpotency class	3
isoclinism	isoclinic class of #330923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 126 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^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, 238131);