#512804

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	#532	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,f^2=g,g^2=h,h^2=i,b^a=be,c^a=ci,d^a=de,c^b=ce,d^b=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420000);